🕷️ Crawler Inspector

URL Lookup

Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 2 (from laksa042)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa002.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.symbolab.com/solver/integral-calculator\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.symbolab.com/solver/integral-calculator\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.symbolab.com\/solver\/integral-calculator","crawl_time":1775402812,"first_indexed_time":1421446798,"http_code":200,"src_unparsed":"com,symbolab!www,\/solver\/integral-calculator s443","src_root_hash":"1977249391401099202","history_drop_reason":null,"meta_title":"Integral Calculator","meta_descriptions":["Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Type in any integral to get the solution, steps and graph"],"attrs_boilerpipe_text":"Introduction to Integration\nWhat is an Integration?\nIntegration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the\ngraph of a function\n. Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form.\nWe distinguish integration into two forms:\ndefinite\nand\nindefinite integrals\n. Fundamental instruments in\ncalculus\n, differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches.\nIntegration - An Inverse Process of Differentiation\nIntegration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one.\nIf  $\\frac{d(F(x))}{dx}=f(x)$, then $\\int f(x)dx=F(x)+C$ . This is known as indefinite integrals.\nFor Example\nSuppose $f(x)=x^3$\nThe\nderivative\nof f(x) is $f'(x)=3x^2$\nThe\nantiderivative\nof $3x^2$ is $x^3$\nSo the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is $ \\int 3x^2dx=x^3+C$\nTherefore, if $\\frac{dy}{dx}=f(x)$, then we can write it as y = $\\int f(x)dx=F(x)+C$ where:\n$\\int f(x)dx$ will represent the complete class of integral.\nC is an arbitrary constant.\nx is the variable of equation.\nThe symbol $\\int$ denotes the integral.\nf(x) is the integrand.\nRules of Integration\nSum and Difference Rules:\n$\\int [f(x) + g(x)] dx = \\int f(x) dx + \\int g(x) dx$\n$\\int [f(x) - g(x)] dx = \\int f(x) dx - \\int g(x) dx$\nFor example:  $\\int (x^2 + 3x) dx = \\int x^2 dx + \\int 3x dx$\n$= \\frac{x^3}{3} + \\frac{3x^2}{2} + C$\nPower Rule:\n$ \\int x^ndx= \\frac {x^{(n+1)}}{n+1} +C $\nPlease note here n$\\neq$-1\nFor example: $\\int x^5dx=\\frac{x^6}{6}+C$\nExponential Rules:\n$ \\int e^xdx=e^x+C$\n$ \\int a^xdx=  \\frac{a^x}{ln(a)}+C$\n$ \\int ln(x)dx=xln(x)-x+C$\nConstant Multiplication Rule:\n$ \\int adx=ax+C $\nReciprocal Rule:\n$  \\int \\frac{1}{x}dx=ln|x|+C $\nProperties of Integration\nProperties of indefinite\n$\\int [f(x) \\pm g(x)]dx= \\int f(x)dx \\pm \\int g(x)dx $\n$ \\int kf(x)dx = k \\int f(x)dx   $ (here k is the constant)\n$ \\int f(x)dx= \\int g(x)dx $ if $ \\int [f(x)-g(x)]dx=0 $\nBy collabrating these properties, we derive: $\\int \\sum k _nf_n(x)dx=\\sum k_n\\int f_n(x)dx $\nWhat are some uses for an integral calculator and what is it exactly?\nOne may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals.\nImplementations of an integral calculator:\nTwo examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative.\nDouble-checking hand computations helps one verify integration answers.\n\"Handling complex functions\" is the capacity to combine activities that are challenging for manual handling.\nApplications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs.\nImproving learning means giving students help understanding integration techniques and their uses.\nFor example using an integration calculator we can find:\n$\\int (3x^2-2x+1)dx\n=3\\frac{x^{2+1}}{2+1}-2 \\frac{x^{1+1}}{1+1}+x+C$\n$=3\\frac{x^{3}}{3}-2 \\frac{x^{2}}{2}+x+C$\n$=x^3-x^2+x+C$\nMethods of Integration\nThe simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here:\n1. Integration by Decomposition\nIn this method we need to breakthe function into basic parts:\n$$\\int \\frac{x^2-x+1}{x^3}dx$$\nExpanding:\n$$\\int (\\frac{x^2}{x^3}-\\frac{x}{x^3}+\\frac{1}{x^3})dx $$\nApplying the basic rules:\n$ log |x|+ \\frac {1}{x}-\\frac{1}{2x^2}+C$\n2. Integration by Substitution\nFor simplyfing the integral change the variables:\n$$ \\int sin(mx)dx $$ Let mx=t, so $dx=dt\/m$\nTherefore, $ \\frac{1}{m} \\int sin t dt= -\\frac{1}{m}cos t +C $\n3. Integration using Partial Fractions\nWe use this function for rational functions: $ \\int \\frac{1}{(x+1)(x+2)}dx=\\frac{A}{x+1}+ \\frac{B}{x+2} $ so now we can integrate them seperately and solve for the value of A and B.\n4. Integration by Parts\nDerived from the product rule of differentiation:\n$$ \\int udv = uv - \\int vdu $$\nApplications of Integration\nIntegration has great use in domains like physics, engineering, and economics. Of the greatest significance are:\nOne may get areas by use of the area under curves.\nThe volume computation is mostly about figuring the solid of rotation volume.\nApplications of physics abound in motion problems, work done, and the center of mass.\nCalculation of probability density functions makes use of statistics and probability.\nThis extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus.\nFrequently Asked Questions (FAQ)\nWhat is the use of integration in real life?\nIntegrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models.\nWhat is the best integral calculator?\nSymbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more.\nWhat does to integrate mean?\nIntegration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.","attrs_markdown":"You can see your coupon in the [user page](https:\/\/www.symbolab.com\/user#subscription)\n\n[Go To QuillBot Upgrade to Pro Continue to site]()\n\n[We've updated our]() [Privacy Policy](https:\/\/www.symbolab.com\/privacy-policy) effective December 15. Please read our updated Privacy Policy and tap\n\nContinue\n\n- [Solutions](https:\/\/www.symbolab.com\/solver)\n  [Integral Calculator](https:\/\/www.symbolab.com\/solver\/integral-calculator) [Derivative Calculator](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [Algebra Calculator](https:\/\/www.symbolab.com\/solver\/algebra-calculator) [Matrix Calculator](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [More...](https:\/\/www.symbolab.com\/solver)\n- [Graphing](https:\/\/www.symbolab.com\/graphing-calculator)\n  [Line Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/line) [Exponential Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/exponential-graph) [Quadratic Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/quadratic-graph) [Sine Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/sine-graph) [More...](https:\/\/www.symbolab.com\/graphing-calculator#examples=true)\n- [Calculators](https:\/\/www.symbolab.com\/calculators)\n  [BMI Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmi) [Compound Interest Calculator](https:\/\/www.symbolab.com\/calculator\/finance\/compound_interest_rate) [Percentage Calculator](https:\/\/www.symbolab.com\/calculator\/math\/percentage) [Acceleration Calculator](https:\/\/www.symbolab.com\/calculator\/physics\/acceleration) [More...](https:\/\/www.symbolab.com\/calculators)\n- [Geometry](https:\/\/www.symbolab.com\/geometry-calculator)\n  [Pythagorean Theorem Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/pythagorean-theorem-calculator) [Circle Area Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/circle-area-diameter-calculator) [Isosceles Triangle Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/isosceles-triangle-prove-angles-calculator) [Triangles Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/triangle-perimeter-calculator) [More...](https:\/\/www.symbolab.com\/geometry-calculator#examples=true)\n- [AI Chat](https:\/\/www.symbolab.com\/ai-chat)\n- [Tools](https:\/\/www.symbolab.com\/solver\/integral-calculator)\n  [Notebook](https:\/\/www.symbolab.com\/notebook)\n  \n  [Groups](https:\/\/www.symbolab.com\/groups)\n  \n  [Cheat Sheets](https:\/\/www.symbolab.com\/cheat-sheets)\n  \n  [Worksheets](https:\/\/www.symbolab.com\/worksheets\/Pre-Algebra)\n  \n  [Study Guides](https:\/\/www.symbolab.com\/study-guides)\n  \n  [Practice](https:\/\/www.symbolab.com\/practice)\n  \n  [Verify Solution](https:\/\/www.symbolab.com\/verify)\n\n- [Upgrade](https:\/\/www.symbolab.com\/solver\/integral-calculator)\n\nClose\n\nAccept Terms\n\nGood job\\!\n\n![Practice](https:\/\/www.symbolab.com\/public\/images\/practice_icon1.png) Practice More\n\nType your Answer\n\n**Verify**\n\n|   |   |   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|   |   |   |   |   |   |   |   |   |   |   |   |\n| x^2 | x^{\\\\msquare} | \\\\log\\_{\\\\msquare} | \\\\sqrt{\\\\square} | \\\\nthroot\\[\\\\msquare\\]{\\\\square} | \\\\le | \\\\ge | \\\\frac{\\\\msquare}{\\\\msquare} | \\\\cdot | \\\\div | x^{\\\\circ} | \\\\pi |\n| \\\\left(\\\\square\\\\right)^{'} | \\\\frac{d}{dx} | \\\\frac{\\\\partial}{\\\\partial x} | \\\\int | \\\\int\\_{\\\\msquare}^{\\\\msquare} | \\\\lim | \\\\sum | \\\\infty | \\\\theta | (f\\\\:\\\\circ\\\\:g) | f(x) | ![](https:\/\/www.symbolab.com\/public\/img\/pad_matrixnxn.png) |\n\n|   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|\n| ▭\\\\:\\\\longdivision{▭} | \\\\times \\\\twostack{▭}{▭} | \\+ \\\\twostack{▭}{▭} | \\- \\\\twostack{▭}{▭} | \\\\left( | \\\\right) | \\\\times | \\\\square\\\\frac{\\\\square}{\\\\square} |\n\nTake a challenge\n\nSubscribe to verify your answer\n\nSubscribe\n\nAre you sure you want to leave this Challenge? By closing this window you will lose this challenge\n\nCancel\n\nLeave\n\n- [Pre Algebra](https:\/\/www.symbolab.com\/solver\/pre-algebra-calculator)\n  [Order of Operations](https:\/\/www.symbolab.com\/solver\/order-of-operations-calculator) [Factors & Primes](https:\/\/www.symbolab.com\/solver\/factors-primes-calculator) [Fractions](https:\/\/www.symbolab.com\/solver\/fractions-calculator) [Long Arithmetic](https:\/\/www.symbolab.com\/solver\/long-arithmetic-calculator) [Decimals](https:\/\/www.symbolab.com\/solver\/decimals-calculator) [Exponents & Radicals](https:\/\/www.symbolab.com\/solver\/exponents-radicals-calculator) [Ratios & Proportions](https:\/\/www.symbolab.com\/solver\/ratios-proportions-calculator) [Percent](https:\/\/www.symbolab.com\/solver\/percent-calculator) [Modulo](https:\/\/www.symbolab.com\/solver\/modulo-calculator) [Number Line](https:\/\/www.symbolab.com\/solver\/number-line-calculator) [Expanded Form](https:\/\/www.symbolab.com\/solver\/expanded-form-calculator) [Mean, Median & Mode](https:\/\/www.symbolab.com\/solver\/mean-median-mode-calculator)\n- [Algebra](https:\/\/www.symbolab.com\/solver\/algebra-calculator)\n  [Equations](https:\/\/www.symbolab.com\/solver\/equation-calculator) [Inequalities](https:\/\/www.symbolab.com\/solver\/inequalities-calculator) [System of Equations](https:\/\/www.symbolab.com\/solver\/system-of-equations-calculator) [System of Inequalities](https:\/\/www.symbolab.com\/solver\/system-of-inequalities-calculator) [Testing Solutions](https:\/\/www.symbolab.com\/solver\/testing-solutions-calculator) [Basic Operations](https:\/\/www.symbolab.com\/solver\/basic-operations-calculator) [Algebraic Properties](https:\/\/www.symbolab.com\/solver\/algebraic-properties-calculator) [Partial Fractions](https:\/\/www.symbolab.com\/solver\/partial-fractions-calculator) [Polynomials](https:\/\/www.symbolab.com\/solver\/polynomial-calculator) [Rational Expressions](https:\/\/www.symbolab.com\/solver\/rational-expression-calculator) [Sequences](https:\/\/www.symbolab.com\/solver\/sequence-calculator) [Power Sums](https:\/\/www.symbolab.com\/solver\/power-sums-calculator) [Interval Notation](https:\/\/www.symbolab.com\/solver\/interval-notation-calculator) [Pi (Product) Notation](https:\/\/www.symbolab.com\/solver\/pi-product-notation-calculator) [Induction](https:\/\/www.symbolab.com\/solver\/induction-calculator) [Prove That](https:\/\/www.symbolab.com\/solver\/prove-that-calculator) [Logical Sets](https:\/\/www.symbolab.com\/solver\/logical-sets-calculator) [Word Problems](https:\/\/www.symbolab.com\/solver\/word-problems-calculator)\n- [Pre Calculus](https:\/\/www.symbolab.com\/solver\/pre-calculus-calculator)\n  [Equations](https:\/\/www.symbolab.com\/solver\/pre-calculus-equation-calculator) [Inequalities](https:\/\/www.symbolab.com\/solver\/pre-calculus-inequalities-calculator) [Scientific Calculator](https:\/\/www.symbolab.com\/solver\/scientific-calculator) [Scientific Notation Arithmetics](https:\/\/www.symbolab.com\/solver\/scientific-notation-arithmetics) [Complex Numbers](https:\/\/www.symbolab.com\/solver\/pre-calculus-complex-numbers-calculator) [Polar\/Cartesian](https:\/\/www.symbolab.com\/solver\/polar-cartesian-calculator) [Simultaneous Equations](https:\/\/www.symbolab.com\/solver\/simultaneous-equations-calculator) [System of Inequalities](https:\/\/www.symbolab.com\/solver\/pre-calculus-system-of-inequalities-calculator) [Polynomials](https:\/\/www.symbolab.com\/solver\/pre-calculus-polynomial-calculator) [Rationales](https:\/\/www.symbolab.com\/solver\/rationales-calculator) [Functions](https:\/\/www.symbolab.com\/solver\/pre-calculus-functions-calculator) [Arithmetic & Comp.](https:\/\/www.symbolab.com\/solver\/pre-calculus-function-arithmetic-composition-calculator) [Coordinate Geometry](https:\/\/www.symbolab.com\/solver\/pre-calculus-coordinate-geometry-calculator) [Plane Geometry](https:\/\/www.symbolab.com\/solver\/plane-geometry-calculator) [Solid Geometry](https:\/\/www.symbolab.com\/solver\/solid-geometry-calculator) [Trigonometry](https:\/\/www.symbolab.com\/solver\/pre-calculus-trigonometry-calculator)\n- [Calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator)\n  [Derivatives](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [Derivative Applications](https:\/\/www.symbolab.com\/solver\/derivative-applications-calculator) [Limits](https:\/\/www.symbolab.com\/solver\/limit-calculator) [Integrals](https:\/\/www.symbolab.com\/solver\/integral-calculator) [Integral Applications](https:\/\/www.symbolab.com\/solver\/integral-applications-calculator) [Integral Approximation](https:\/\/www.symbolab.com\/solver\/integral-approximation-calculator) [Series](https:\/\/www.symbolab.com\/solver\/series-calculator) [ODE](https:\/\/www.symbolab.com\/solver\/ordinary-differential-equation-calculator) [Multivariable Calculus](https:\/\/www.symbolab.com\/solver\/multivariable-calculus-calculator) [Laplace Transform](https:\/\/www.symbolab.com\/solver\/laplace-calculator) [Taylor\/Maclaurin Series](https:\/\/www.symbolab.com\/solver\/taylor-maclaurin-series-calculator) [Fourier Series](https:\/\/www.symbolab.com\/solver\/fourier-series-calculator) [Fourier Transform](https:\/\/www.symbolab.com\/solver\/fourier-transform-calculator)\n- [Functions](https:\/\/www.symbolab.com\/solver\/functions-line-calculator)\n  [Line Equations](https:\/\/www.symbolab.com\/solver\/line-equation-calculator) [Functions](https:\/\/www.symbolab.com\/solver\/functions-calculator) [Arithmetic & Comp.](https:\/\/www.symbolab.com\/solver\/function-arithmetic-composition-calculator) [Conic Sections](https:\/\/www.symbolab.com\/solver\/conic-sections-calculator) [Transformation](https:\/\/www.symbolab.com\/solver\/function-transformation-calculator)\n- [Linear Algebra](https:\/\/www.symbolab.com\/solver\/linear-algebra-calculator)\n  [Matrices](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [Vectors](https:\/\/www.symbolab.com\/solver\/vector-calculator)\n- [Trigonometry](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator)\n  [Quadrant](https:\/\/www.symbolab.com\/solver\/quadrant-calculator) [Coterminal Angle](https:\/\/www.symbolab.com\/solver\/coterminal-angle-calculator) [Identities](https:\/\/www.symbolab.com\/solver\/trigonometric-identity-calculator) [Proving Identities](https:\/\/www.symbolab.com\/solver\/trigonometric-identity-proving-calculator) [Trig Equations](https:\/\/www.symbolab.com\/solver\/trigonometric-equation-calculator) [Trig Inequalities](https:\/\/www.symbolab.com\/solver\/trigonometric-inequality-calculator) [Evaluate Functions](https:\/\/www.symbolab.com\/solver\/trigonometric-function-calculator) [Simplify](https:\/\/www.symbolab.com\/solver\/trigonometric-simplification-calculator)\n- [Statistics](https:\/\/www.symbolab.com\/solver\/statistics-calculator)\n  [Mean](https:\/\/www.symbolab.com\/solver\/arithmetic-mean-calculator) [Geometric Mean](https:\/\/www.symbolab.com\/solver\/geometric-mean-calculator) [Quadratic Mean](https:\/\/www.symbolab.com\/solver\/quadratic-mean-calculator) [Average](https:\/\/www.symbolab.com\/solver\/average-calculator) [Median](https:\/\/www.symbolab.com\/solver\/median-calculator) [Mode](https:\/\/www.symbolab.com\/solver\/mode-calculator) [Order](https:\/\/www.symbolab.com\/solver\/order-calculator) [Minimum](https:\/\/www.symbolab.com\/solver\/minimum-calculator) [Maximum](https:\/\/www.symbolab.com\/solver\/maximum-calculator) [Probability](https:\/\/www.symbolab.com\/solver\/probability-calculator) [Mid-Range](https:\/\/www.symbolab.com\/solver\/mid-range-calculator) [Range](https:\/\/www.symbolab.com\/solver\/range-calculator) [Standard Deviation](https:\/\/www.symbolab.com\/solver\/standard-deviation-calculator) [Variance](https:\/\/www.symbolab.com\/solver\/variance-calculator) [Lower Quartile](https:\/\/www.symbolab.com\/solver\/lower-quartile-calculator) [Upper Quartile](https:\/\/www.symbolab.com\/solver\/upper-quartile-calculator) [Interquartile Range](https:\/\/www.symbolab.com\/solver\/interquartile-range-calculator) [Midhinge](https:\/\/www.symbolab.com\/solver\/midhinge-calculator) [Standard Normal Distribution](https:\/\/www.symbolab.com\/solver\/standard-normal-distribution-calculator)\n- [Physics](https:\/\/www.symbolab.com\/solver\/physics-calculator)\n  [Mechanics](https:\/\/www.symbolab.com\/solver\/mechanics-calculator)\n- [Chemistry](https:\/\/www.symbolab.com\/solver\/chemistry-calculator)\n  [Chemical Reactions](https:\/\/www.symbolab.com\/solver\/chemical-reaction-calculator) [Chemical Properties](https:\/\/www.symbolab.com\/solver\/chemical-properties-calculator)\n- [Finance](https:\/\/www.symbolab.com\/solver\/finance-calculator)\n  [Simple Interest](https:\/\/www.symbolab.com\/solver\/simple-interest-calculator) [Compound Interest](https:\/\/www.symbolab.com\/solver\/compound-interest-calculator) [Present Value](https:\/\/www.symbolab.com\/solver\/present-value-calculator) [Future Value](https:\/\/www.symbolab.com\/solver\/future-value-calculator)\n- [Economics](https:\/\/www.symbolab.com\/solver\/economics-calculator)\n  [Point of Diminishing Return](https:\/\/www.symbolab.com\/solver\/point-of-diminishing-return-calculator)\n- [Conversions](https:\/\/www.symbolab.com\/solver\/conversion-calculator)\n  [Currency](https:\/\/www.symbolab.com\/solver\/currency-converter) [Roman Numerals](https:\/\/www.symbolab.com\/solver\/roman-numeral-converter) [Radical to Exponent](https:\/\/www.symbolab.com\/solver\/radical-to-exponent-calculator) [Exponent to Radical](https:\/\/www.symbolab.com\/solver\/exponent-to-radical-calculator) [To Fraction](https:\/\/www.symbolab.com\/solver\/convert-to-fraction-calculator) [To Decimal](https:\/\/www.symbolab.com\/solver\/convert-to-decimal-calculator) [To Mixed Number](https:\/\/www.symbolab.com\/solver\/convert-to-mixed-number-calculator) [To Improper Fraction](https:\/\/www.symbolab.com\/solver\/convert-to-improper-fraction-calculator) [Radians to Degrees](https:\/\/www.symbolab.com\/solver\/radians-to-degrees-calculator) [Degrees to Radians](https:\/\/www.symbolab.com\/solver\/degrees-to-radians-calculator)  [Degrees Minutes Seconds](https:\/\/www.symbolab.com\/solver\/degrees-minutes-seconds-calculator) [Hexadecimal](https:\/\/www.symbolab.com\/solver\/hexadecimal-converter) [Scientific Notation](https:\/\/www.symbolab.com\/solver\/scientific-notation-converter) [Distance](https:\/\/www.symbolab.com\/solver\/distance-converter) [Weight](https:\/\/www.symbolab.com\/solver\/weight-converter) [Time](https:\/\/www.symbolab.com\/solver\/time-converter) [Volume](https:\/\/www.symbolab.com\/solver\/volume-converter)\n\n### Special Offer\n## Double the Tools. One Smart Bundle.\nto master your math and writing\\!\n\nPro\n\n- Full access to solution steps\n- Full access to AI chat\n- Practice and improve\n- Access from any device\n- No ads\n- Unlimited storage\n\n- Detect AI-generated content\n- Advanced Grammar Checker\n- AI Humanizer\n- Paraphrase in unlimited modes\n- Summarize any text\n- Much More\\!\n\n[Restrictions apply, check out our FAQs.](https:\/\/www.symbolab.com\/bundle-faq)\n\nGo back\n\nPurchase Bundle\n\n### Back to School Promotion\nAnnual\n\nAnnual - \\$\n\n%\n\nOFF\n\nAnnual plan\n\nOne time offer\n\nfor one year, then \\$\n\nGo back\n\n\\+\n\n## qb-banner-title\n###\n\n##\n###\n\n[Solutions](https:\/\/www.symbolab.com\/solver) \\> [Calculus Calculator](https:\/\/www.symbolab.com\/solver\/calculus-calculator) \\>\n\n# Integral Calculator\n\nTopic\n\n- [Pre Algebra](https:\/\/www.symbolab.com\/solver\/pre-algebra-calculator)\n- [Algebra](https:\/\/www.symbolab.com\/solver\/algebra-calculator)\n- [Pre Calculus](https:\/\/www.symbolab.com\/solver\/pre-calculus-calculator)\n- [Calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator)\n- [Derivatives]()\n  - [First Derivative](https:\/\/www.symbolab.com\/solver\/first-derivative-calculator)\n  - [WRT](https:\/\/www.symbolab.com\/solver\/derivative-with-respect-calculator)\n  - [Specify Method]()\n    - [Chain Rule](https:\/\/www.symbolab.com\/solver\/derivative-chain-rule-calculator)\n    - [Product Rule](https:\/\/www.symbolab.com\/solver\/derivative-product-rule-calculator)\n    - [Quotient Rule](https:\/\/www.symbolab.com\/solver\/derivative-quotient-rule-calculator)\n    - [Sum\/Diff Rule](https:\/\/www.symbolab.com\/solver\/derivative-sum-diff-rule-calculator)\n  - [Second Derivative](https:\/\/www.symbolab.com\/solver\/second-derivative-calculator)\n  - [Third Derivative](https:\/\/www.symbolab.com\/solver\/third-derivative-calculator)\n  - [Higher Order Derivatives](https:\/\/www.symbolab.com\/solver\/higher-order-derivative-calculator)\n  - [Derivative at a point](https:\/\/www.symbolab.com\/solver\/derivative-point-calculator)\n  - [Partial Derivative](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n  - [Implicit Derivative](https:\/\/www.symbolab.com\/solver\/implicit-derivative-calculator)\n  - [Second Implicit Derivative](https:\/\/www.symbolab.com\/solver\/second-implicit-derivative-calculator)\n  - [Derivative using Definition](https:\/\/www.symbolab.com\/solver\/derivative-using-definition-calculator)\n- [Derivative Applications]()\n  - [Tangent](https:\/\/www.symbolab.com\/solver\/tangent-line-calculator)\n  - [Slope of Tangent](https:\/\/www.symbolab.com\/solver\/slope-of-tangent-calculator)\n  - [Normal](https:\/\/www.symbolab.com\/solver\/normal-line-calculator)\n  - [Curved Line Slope](https:\/\/www.symbolab.com\/solver\/curved-line-slope-calculator)\n  - [Extreme Points](https:\/\/www.symbolab.com\/solver\/calculus-function-extreme-points-calculator)\n  - [Tangent to Conic](https:\/\/www.symbolab.com\/solver\/tangent-to-conic-calculator)\n  - [Linear Approximation](https:\/\/www.symbolab.com\/solver\/linear-approximation-calculator)\n  - [Difference Quotient](https:\/\/www.symbolab.com\/solver\/difference-quotient-calculator)\n  - [Horizontal Tangent](https:\/\/www.symbolab.com\/solver\/horizontal-tangent-calculator)\n- [Limits]()\n  - [One Variable](https:\/\/www.symbolab.com\/solver\/single-var-limit-calculator)\n  - [Multi Variable Limit](https:\/\/www.symbolab.com\/solver\/multi-var-limit-calculator)\n  - [One Sided](https:\/\/www.symbolab.com\/solver\/one-sided-limit-calculator)\n  - [At Infinity](https:\/\/www.symbolab.com\/solver\/limit-infinity-calculator)\n  - [Specify Method]()\n    - [L'Hopital's Rule](https:\/\/www.symbolab.com\/solver\/limit-lhopital-calculator)\n    - [Squeeze Theorem](https:\/\/www.symbolab.com\/solver\/limit-squeeze-theorem-calculator)\n    - [Chain Rule](https:\/\/www.symbolab.com\/solver\/limit-chain-rule-calculator)\n    - [Factoring](https:\/\/www.symbolab.com\/solver\/limit-using-factoring-calculator)\n    - [Substitution](https:\/\/www.symbolab.com\/solver\/limit-substitution-calculator)\n    - [Sandwich Theorem](https:\/\/www.symbolab.com\/solver\/limit-sandwich-theorem-calculator)\n- [Integrals]()\n  - [Indefinite Integrals](https:\/\/www.symbolab.com\/solver\/indefinite-integral-calculator)\n  - [Definite Integrals](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator)\n  - [Specific-Method]()\n    - [Partial Fractions](https:\/\/www.symbolab.com\/solver\/partial-fractions-integration-calculator)\n    - [U-Substitution](https:\/\/www.symbolab.com\/solver\/u-substitution-integration-calculator)\n    - [Trigonometric Substitution](https:\/\/www.symbolab.com\/solver\/trigonometric-substitution-integration-calculator)\n    - [Weierstrass Substitution](https:\/\/www.symbolab.com\/solver\/weierstrass-substitution-calculator)\n    - [By Parts](https:\/\/www.symbolab.com\/solver\/by-parts-integration-calculator)\n    - [Long Division](https:\/\/www.symbolab.com\/solver\/long-division-integration-calculator)\n  - [Improper Integrals](https:\/\/www.symbolab.com\/solver\/improper-integral-calculator)\n  - [Antiderivatives](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator)\n  - [Double Integrals](https:\/\/www.symbolab.com\/solver\/double-integrals-calculator)\n  - [Triple Integrals](https:\/\/www.symbolab.com\/solver\/triple-integrals-calculator)\n  - [Multiple Integrals](https:\/\/www.symbolab.com\/solver\/multiple-integrals-calculator)\n- [Integral Applications]()\n  - [Limit of Sum](https:\/\/www.symbolab.com\/solver\/limit-of-sum-calculator)\n  - [Area under curve](https:\/\/www.symbolab.com\/solver\/area-under-curve-calculator)\n  - [Area between curves](https:\/\/www.symbolab.com\/solver\/area-between-curves-calculator)\n  - [Area under polar curve](https:\/\/www.symbolab.com\/solver\/area-under-polar-curve-calculator)\n  - [Volume of solid of revolution](https:\/\/www.symbolab.com\/solver\/volume-calculator)\n  - [Arc Length](https:\/\/www.symbolab.com\/solver\/arc-length-calculator)\n  - [Function Average](https:\/\/www.symbolab.com\/solver\/function-average-calculator)\n- [Integral Approximation]()\n  - [Riemann Sum](https:\/\/www.symbolab.com\/solver\/riemann-sum-calculator)\n  - [Trapezoidal](https:\/\/www.symbolab.com\/solver\/trapezoidal-approximation-calculator)\n  - [Simpson's Rule](https:\/\/www.symbolab.com\/solver\/simpsons-rule-calculator)\n  - [Midpoint Rule](https:\/\/www.symbolab.com\/solver\/midpoint-rule-calculator)\n- [Series]()\n  - [Convergence]()\n    - [Geometric Series Test](https:\/\/www.symbolab.com\/solver\/geometric-series-test-calculator)\n    - [Telescoping Series Test](https:\/\/www.symbolab.com\/solver\/telescoping-series-test-calculator)\n    - [Alternating Series Test](https:\/\/www.symbolab.com\/solver\/alternating-series-test-calculator)\n    - [P Series Test](https:\/\/www.symbolab.com\/solver\/p-series-test-calculator)\n    - [Divergence Test](https:\/\/www.symbolab.com\/solver\/series-divergence-test-calculator)\n    - [Ratio Test](https:\/\/www.symbolab.com\/solver\/series-ratio-test-calculator)\n    - [Root Test](https:\/\/www.symbolab.com\/solver\/series-root-test-calculator)\n    - [Comparison Test](https:\/\/www.symbolab.com\/solver\/series-comparison-test-calculator)\n    - [Limit Comparison Test](https:\/\/www.symbolab.com\/solver\/series-limit-comparison-test-calculator)\n    - [Integral Test](https:\/\/www.symbolab.com\/solver\/series-integral-test-calculator)\n  - [Absolute Convergence](https:\/\/www.symbolab.com\/solver\/series-absolute-convergence-calculator)\n  - [Power Series]()\n    - [Radius of Convergence](https:\/\/www.symbolab.com\/solver\/radius-of-convergence-calculator)\n    - [Interval of Convergence](https:\/\/www.symbolab.com\/solver\/interval-of-convergence-calculator)\n- [ODE]()\n  - [Linear First Order](https:\/\/www.symbolab.com\/solver\/linear-first-order-differential-equation-calculator)\n  - [Linear w\/constant coefficients](https:\/\/www.symbolab.com\/solver\/linear-constant-coefficients-differential-equation-calculator)\n  - [Separable](https:\/\/www.symbolab.com\/solver\/separable-differential-equation-calculator)\n  - [Bernoulli](https:\/\/www.symbolab.com\/solver\/bernoulli-differential-equation-calculator)\n  - [Exact](https:\/\/www.symbolab.com\/solver\/exact-differential-equation-calculator)\n  - [Second Order](https:\/\/www.symbolab.com\/solver\/second-order-differential-equation-calculator)\n  - [Homogenous](https:\/\/www.symbolab.com\/solver\/homogenous-differential-equation-calculator)\n  - [Non Homogenous](https:\/\/www.symbolab.com\/solver\/non-homogenous-differential-equation-calculator)\n  - [Substitution](https:\/\/www.symbolab.com\/solver\/substitution-differential-equation-calculator)\n  - [System of ODEs](https:\/\/www.symbolab.com\/solver\/system-of-differential-equations-calculator)\n  - [IVP using Laplace](https:\/\/www.symbolab.com\/solver\/ivp-laplace-calculator)\n  - [Series Solutions](https:\/\/www.symbolab.com\/solver\/ode-series-solutions-calculator)\n  - [Method of Frobenius](https:\/\/www.symbolab.com\/solver\/frobenius-method-calculator)\n  - [Gamma Function](https:\/\/www.symbolab.com\/solver\/gamma-function-calculator)\n- [Multivariable Calculus]()\n  - [Partial Derivative](https:\/\/www.symbolab.com\/solver\/multivariable-partial-derivative-calculator)\n  - [Implicit Derivative](https:\/\/www.symbolab.com\/solver\/multivariable-implicit-derivative-calculator)\n  - [Tangent to Conic](https:\/\/www.symbolab.com\/solver\/multivariable-tangent-to-conic-calculator)\n  - [Multi Variable Limit](https:\/\/www.symbolab.com\/solver\/multi-var-calculus-limit-calculator)\n  - [Multiple Integrals](https:\/\/www.symbolab.com\/solver\/multiple-integrals-multivariable-calculator)\n  - [Gradient](https:\/\/www.symbolab.com\/solver\/gradient-calculator)\n  - [Divergence](https:\/\/www.symbolab.com\/solver\/divergence-calculator)\n  - [Extreme Points](https:\/\/www.symbolab.com\/solver\/multi-var-function-extreme-points-calculator)\n- [Laplace Transform]()\n  - [Inverse](https:\/\/www.symbolab.com\/solver\/inverse-laplace-calculator)\n- [Taylor\/Maclaurin Series]()\n  - [Taylor Series](https:\/\/www.symbolab.com\/solver\/taylor-series-calculator)\n  - [Maclaurin Series](https:\/\/www.symbolab.com\/solver\/maclaurin-series-calculator)\n- [Fourier Series](https:\/\/www.symbolab.com\/solver\/fourier-series-calculator)\n- [Fourier Transform](https:\/\/www.symbolab.com\/solver\/fourier-transform-calculator)\n- [Functions](https:\/\/www.symbolab.com\/solver\/functions-line-calculator)\n- [Linear Algebra](https:\/\/www.symbolab.com\/solver\/linear-algebra-calculator)\n- [Trigonometry](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator)\n- [Statistics](https:\/\/www.symbolab.com\/solver\/statistics-calculator)\n- [Physics](https:\/\/www.symbolab.com\/solver\/physics-calculator)\n- [Chemistry](https:\/\/www.symbolab.com\/solver\/chemistry-calculator)\n- [Finance](https:\/\/www.symbolab.com\/solver\/finance-calculator)\n- [Economics](https:\/\/www.symbolab.com\/solver\/economics-calculator)\n- [Conversions](https:\/\/www.symbolab.com\/solver\/conversion-calculator)\n\nAdd to Chrome\n\nGet our extension, you can capture any math problem from any website\n\nFull pad\n\n|   |   |   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|   |   |   |   |   |   |   |   |   |   |   |   |\n| x^2 | x^{\\\\msquare} | \\\\log\\_{\\\\msquare} | \\\\sqrt{\\\\square} | \\\\nthroot\\[\\\\msquare\\]{\\\\square} | \\\\le | \\\\ge | \\\\frac{\\\\msquare}{\\\\msquare} | \\\\cdot | \\\\div | x^{\\\\circ} | \\\\pi |\n| \\\\left(\\\\square\\\\right)^{'} | \\\\frac{d}{dx} | \\\\frac{\\\\partial}{\\\\partial x} | \\\\int | \\\\int\\_{\\\\msquare}^{\\\\msquare} | \\\\lim | \\\\sum | \\\\infty | \\\\theta | (f\\\\:\\\\circ\\\\:g) | f(x) | ![](https:\/\/www.symbolab.com\/public\/img\/pad_matrixnxn.png) |\n\n|   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|\n| \\- \\\\twostack{▭}{▭} | \\\\lt | 7 | 8 | 9 | \\\\div | AC |\n| \\+ \\\\twostack{▭}{▭} | \\\\gt | 4 | 5 | 6 | \\\\times | \\\\square\\\\frac{\\\\square}{\\\\square} |\n| \\\\times \\\\twostack{▭}{▭} | \\\\left( | 1 | 2 | 3 | \\- | x |\n| ▭\\\\:\\\\longdivision{▭} | \\\\right) | . | 0 | \\= | \\+ | y |\n\n\\\\mathrm{partial\\\\:fractions}    \\\\mathrm{substitution}    \\\\mathrm{long\\\\:division}    \\\\mathrm{trigonometric\\\\:substitution}    \\\\mathrm{by\\\\:parts}\n\nSee All\n\narea\n\nasymptotes\n\ncritical points\n\nderivative\n\ndomain\n\neigenvalues\n\neigenvectors\n\nexpand\n\nextreme points\n\nfactor\n\nimplicit derivative\n\ninflection points\n\nintercepts\n\ninverse\n\nlaplace\n\ninverse laplace\n\npartial fractions\n\nrange\n\nslope\n\nsimplify\n\nsolve for\n\ntangent\n\ntaylor\n\nvertex\n\ngeometric test\n\nalternating test\n\ntelescoping test\n\npseries test\n\nroot test\n\n**Go**\n\n[Steps](https:\/\/www.symbolab.com\/solver\/integral-calculator#steps) [Graph](https:\/\/www.symbolab.com\/solver\/integral-calculator#plot) [Related](https:\/\/www.symbolab.com\/solver\/integral-calculator#related_anchor) [Examples](https:\/\/www.symbolab.com\/solver\/integral-calculator#Examples)\n\nGenerated by AI\n\nAI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view.\n\nVerify your Answer\n\nSubscribe to verify your answer\n\nSubscribe\n\nSave to Notebook\\!\n\nSign in to save notes\n\nSign in\n\nVerify\n\nSave\n\nShow Steps\n\nHide Steps\n\n## Number Line\n![]()\n\n![](https:\/\/www.symbolab.com\/public\/img\/ajax-loader.gif)\n\nRelated\n\nIntegral Examples\n\n- [\\\\int e^x\\\\cos(x)dx](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint%20e%5Ex%5Ccos%28x%29dx?or=ex)\n- [\\\\int \\\\cos^3(x)\\\\sin (x)dx](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint%20%5Ccos%5E3%28x%29%5Csin%20%28x%29dx?or=ex)\n- [\\\\int \\\\frac{2x+1}{(x+5)^3}](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint%20%5Cfrac%7B2x%2B1%7D%7B%28x%2B5%29%5E3%7D?or=ex)\n- [\\\\int\\_{0}^{\\\\pi}\\\\sin(x)dx](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Csin%28x%29dx?or=ex)\n- [\\\\int\\_{a}^{b} x^2dx](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint_%7Ba%7D%5E%7Bb%7D%20x%5E2dx?or=ex)\n- [\\\\int\\_{0}^{2\\\\pi}\\\\cos^2(\\\\theta)d\\\\theta](https:\/\/www.symbolab.com\/solver\/integral-calculator\/%5Cint_%7B0%7D%5E%7B2%5Cpi%7D%5Ccos%5E2%28%5Ctheta%29d%5Ctheta?or=ex)\n- [partial\\\\:fractions\\\\:\\\\int\\_{0}^{1} \\\\frac{32}{x^{2}-64}dx](https:\/\/www.symbolab.com\/solver\/integral-calculator\/partial%20fractions%20%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%7B32%7D%7Bx%5E%7B2%7D-64%7Ddx?or=ex)\n- [substitution\\\\:\\\\int\\\\frac{e^{x}}{e^{x}+e^{-x}}dx,\\\\:u=e^{x}](https:\/\/www.symbolab.com\/solver\/integral-calculator\/substitution%20%5Cint%5Cfrac%7Be%5E%7Bx%7D%7D%7Be%5E%7Bx%7D%2Be%5E%7B-x%7D%7Ddx%2C%20u%3De%5E%7Bx%7D?or=ex)\n- Show More\n\n## Introduction to Integration\n### What is an Integration?\nIntegration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the [graph of a function](https:\/\/www.symbolab.com\/graphing-calculator). Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form.\n\nWe distinguish integration into two forms: [definite](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator) and [indefinite integrals](https:\/\/www.symbolab.com\/solver\/indefinite-integral-calculator). Fundamental instruments in [calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator), differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches.\n\n### Integration - An Inverse Process of Differentiation\nIntegration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one.\n\nIf \\$\\\\frac{d(F(x))}{dx}=f(x)\\$, then \\$\\\\int f(x)dx=F(x)+C\\$ . This is known as indefinite integrals.\n\n**For Example**\n\nSuppose \\$f(x)=x^3\\$\n\nThe [derivative](https:\/\/www.symbolab.com\/solver\/derivative-calculator) of f(x) is \\$f'(x)=3x^2\\$\n\nThe [antiderivative](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator) of \\$3x^2\\$ is \\$x^3\\$\n\nSo the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is \\$ \\\\int 3x^2dx=x^3+C\\$\n\nTherefore, if \\$\\\\frac{dy}{dx}=f(x)\\$, then we can write it as y = \\$\\\\int f(x)dx=F(x)+C\\$ where:\n\n- \\$\\\\int f(x)dx\\$ will represent the complete class of integral.\n- C is an arbitrary constant.\n- x is the variable of equation.\n- The symbol \\$\\\\int\\$ denotes the integral.\n- f(x) is the integrand.\n### Rules of Integration\n**Sum and Difference Rules:**\n\n\\$\\\\int \\[f(x) + g(x)\\] dx = \\\\int f(x) dx + \\\\int g(x) dx\\$\n\n\\$\\\\int \\[f(x) - g(x)\\] dx = \\\\int f(x) dx - \\\\int g(x) dx\\$\n\nFor example: \\$\\\\int (x^2 + 3x) dx = \\\\int x^2 dx + \\\\int 3x dx\\$\n\n\\$= \\\\frac{x^3}{3} + \\\\frac{3x^2}{2} + C\\$\n\n**Power Rule:**\n\n\\$ \\\\int x^ndx= \\\\frac {x^{(n+1)}}{n+1} +C \\$\n\nPlease note here n\\$\\\\neq\\$-1\n\nFor example: \\$\\\\int x^5dx=\\\\frac{x^6}{6}+C\\$\n\n**Exponential Rules:**\n\n\\$ \\\\int e^xdx=e^x+C\\$\n\n\\$ \\\\int a^xdx= \\\\frac{a^x}{ln(a)}+C\\$\n\n\\$ \\\\int ln(x)dx=xln(x)-x+C\\$\n\n**Constant Multiplication Rule:**\n\n\\$ \\\\int adx=ax+C \\$\n\n**Reciprocal Rule:**\n\n\\$ \\\\int \\\\frac{1}{x}dx=ln\\|x\\|+C \\$\n\n### Properties of Integration\nProperties of indefinite\n\n\\$\\\\int \\[f(x) \\\\pm g(x)\\]dx= \\\\int f(x)dx \\\\pm \\\\int g(x)dx \\$\n\n\\$ \\\\int kf(x)dx = k \\\\int f(x)dx \\$ (here k is the constant)\n\n\\$ \\\\int f(x)dx= \\\\int g(x)dx \\$ if \\$ \\\\int \\[f(x)-g(x)\\]dx=0 \\$\n\nBy collabrating these properties, we derive: \\$\\\\int \\\\sum k \\_nf\\_n(x)dx=\\\\sum k\\_n\\\\int f\\_n(x)dx \\$\n\n### What are some uses for an integral calculator and what is it exactly?\nOne may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals.\n\n### Implementations of an integral calculator:\n1. Two examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative.\n2. Double-checking hand computations helps one verify integration answers.\n3. \"Handling complex functions\" is the capacity to combine activities that are challenging for manual handling.\n4. Applications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs.\n5. Improving learning means giving students help understanding integration techniques and their uses.\n\nFor example using an integration calculator we can find:\n\n\\$\\\\int (3x^2-2x+1)dx =3\\\\frac{x^{2+1}}{2+1}-2 \\\\frac{x^{1+1}}{1+1}+x+C\\$ \\$=3\\\\frac{x^{3}}{3}-2 \\\\frac{x^{2}}{2}+x+C\\$ \\$=x^3-x^2+x+C\\$\n\n### Methods of Integration\nThe simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here:\n\n**1\\. Integration by Decomposition**\n\nIn this method we need to breakthe function into basic parts: \\$\\$\\\\int \\\\frac{x^2-x+1}{x^3}dx\\$\\$ Expanding: \\$\\$\\\\int (\\\\frac{x^2}{x^3}-\\\\frac{x}{x^3}+\\\\frac{1}{x^3})dx \\$\\$ Applying the basic rules: \\$ log \\|x\\|+ \\\\frac {1}{x}-\\\\frac{1}{2x^2}+C\\$\n\n**2\\. Integration by Substitution**\n\nFor simplyfing the integral change the variables: \\$\\$ \\\\int sin(mx)dx \\$\\$ Let mx=t, so \\$dx=dt\/m\\$ Therefore, \\$ \\\\frac{1}{m} \\\\int sin t dt= -\\\\frac{1}{m}cos t +C \\$\n\n**3\\. Integration using Partial Fractions**\n\nWe use this function for rational functions: \\$ \\\\int \\\\frac{1}{(x+1)(x+2)}dx=\\\\frac{A}{x+1}+ \\\\frac{B}{x+2} \\$ so now we can integrate them seperately and solve for the value of A and B.\n\n**4\\. Integration by Parts**\n\nDerived from the product rule of differentiation: \\$\\$ \\\\int udv = uv - \\\\int vdu \\$\\$\n\n### Applications of Integration\nIntegration has great use in domains like physics, engineering, and economics. Of the greatest significance are:\n\n1. One may get areas by use of the area under curves.\n2. The volume computation is mostly about figuring the solid of rotation volume.\n3. Applications of physics abound in motion problems, work done, and the center of mass.\n4. Calculation of probability density functions makes use of statistics and probability.\n\nThis extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus.\n\nFrequently Asked Questions (FAQ)\n\n- What is the use of integration in real life?\n- Integrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models.\n- What is the best integral calculator?\n- Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more.\n- What does to integrate mean?\n- Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.\n\nWhy users love our Integral Calculator\n\n|   |   |\n|---|---|\n| 🌐 Languages | EN, ES, PT & more |\n| 🏆 Practice | Improve your math skills |\n| 😍 Step by step | In depth solution steps |\n| ⭐️ Rating | [4\\.6](https:\/\/apps.apple.com\/ua\/app\/symbolab-math-problem-solver\/id876942533)  based on 20924 reviews |\n\nintegral-calculator\n\nen\n\nRelated Symbolab blog posts\n\n[Advanced Math Solutions – Integral Calculator, the complete guide We’ve covered quite a few integration techniques, some are straightforward, some are more challenging, but finding...](https:\/\/blog.symbolab.com\/2015\/01\/advanced-math-solutions-integral_27.html)\n\nPopular topics\n\n[scientific calculator](https:\/\/www.symbolab.com\/solver\/scientific-calculator) [inverse calculator](https:\/\/www.symbolab.com\/solver\/function-inverse-calculator) [simplify calculator](https:\/\/www.symbolab.com\/solver\/simplify-calculator) [distance calculator](https:\/\/www.symbolab.com\/solver\/distance-calculator) [fractions calculator](https:\/\/www.symbolab.com\/solver\/fractions-calculator) [interval notation calculator](https:\/\/www.symbolab.com\/solver\/interval-notation-calculator) [cross product calculator](https:\/\/www.symbolab.com\/solver\/vector-cross-product-calculator) [probability calculator](https:\/\/www.symbolab.com\/solver\/probability-calculator) [derivative calculator](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [series calculator](https:\/\/www.symbolab.com\/solver\/series-calculator) [ratios calculator](https:\/\/www.symbolab.com\/solver\/ratios-calculator) [statistics calculator](https:\/\/www.symbolab.com\/solver\/statistics-calculator) [integral calculator](https:\/\/www.symbolab.com\/solver\/integral-calculator) [inverse laplace transform calculator](https:\/\/www.symbolab.com\/solver\/inverse-laplace-calculator) [rounding calculator](https:\/\/www.symbolab.com\/solver\/rounding-calculator) [gcf calculator](https:\/\/www.symbolab.com\/solver\/gcf-calculator) [algebra calculator](https:\/\/www.symbolab.com\/solver\/algebra-calculator) [tangent line calculator](https:\/\/www.symbolab.com\/solver\/tangent-line-calculator) [trigonometry calculator](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator) [log calculator](https:\/\/www.symbolab.com\/solver\/logarithmic-equation-calculator) [standard deviation calculator](https:\/\/www.symbolab.com\/solver\/standard-deviation-calculator) [linear equation calculator](https:\/\/www.symbolab.com\/solver\/linear-equation-calculator) [antiderivative calculator](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator) [laplace transform calculator](https:\/\/www.symbolab.com\/solver\/laplace-calculator) [quadratic equation calculator](https:\/\/www.symbolab.com\/solver\/quadratic-equation-calculator) [domain calculator](https:\/\/www.symbolab.com\/solver\/function-domain-calculator) [decimals calculator](https:\/\/www.symbolab.com\/solver\/decimals-calculator) [limit calculator](https:\/\/www.symbolab.com\/solver\/limit-calculator) [equation solver](https:\/\/www.symbolab.com\/solver\/equation-calculator) [definite integral calculator](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator) [matrix inverse calculator](https:\/\/www.symbolab.com\/solver\/matrix-inverse-calculator) [matrix calculator](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [system of equations calculator](https:\/\/www.symbolab.com\/solver\/system-of-equations-calculator) [calculus calculator](https:\/\/www.symbolab.com\/solver\/calculus-calculator) [slope calculator](https:\/\/www.symbolab.com\/solver\/slope-calculator) [long division calculator](https:\/\/www.symbolab.com\/solver\/polynomial-long-division-calculator) [factors calculator](https:\/\/www.symbolab.com\/solver\/factor-calculator) [polynomial calculator](https:\/\/www.symbolab.com\/solver\/polynomial-equation-calculator) [square root calculator](https:\/\/www.symbolab.com\/solver\/square-root-calculator) [implicit differentiation calculator](https:\/\/www.symbolab.com\/solver\/implicit-derivative-calculator) [word problem solver](https:\/\/www.symbolab.com\/solver\/word-problems-calculator) [differential equation calculator](https:\/\/www.symbolab.com\/solver\/ordinary-differential-equation-calculator) [average calculator](https:\/\/www.symbolab.com\/calculator\/statistics\/average) [synthetic division calculator](https:\/\/www.symbolab.com\/solver\/synthetic-division-calculator)\n\nChat with Symbo AI\n\nAI may present inaccurate or offensive content that does not represent Symbolab's views.\n\nDo not enter any personal information\n\nEnter a problem\n\n[Cooking Calculators](https:\/\/www.symbolab.com\/calculator\/cooking)\n\n[Cooking Measurement Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cooking-measurement) [Cooking Ingredient Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cooking-ingredient) [Cake Pan Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cake-pan) [More calculators](https:\/\/www.symbolab.com\/calculator\/cooking)\n\n[Fitness Calculators](https:\/\/www.symbolab.com\/calculator\/fitness)\n\n[BMI Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmi) [Calorie Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/calorie) [BMR Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmr) [More calculators](https:\/\/www.symbolab.com\/calculator\/fitness)\n\nSave to Notebook\\!\n\nSign in\n\nNotebook\n\n[View Full Notebook](https:\/\/www.symbolab.com\/notebook)\n\nStudy Tools [AI Math Solver](https:\/\/www.symbolab.com\/) [AI Chat](https:\/\/www.symbolab.com\/ai-chat) [Popular Problems](https:\/\/www.symbolab.com\/popular-algebra) [Worksheets](https:\/\/www.symbolab.com\/worksheets\/Pre-Algebra) [Study Guides](https:\/\/www.symbolab.com\/study-guides) [Practice](https:\/\/www.symbolab.com\/practice) [Cheat Sheets](https:\/\/www.symbolab.com\/cheat-sheets) [Calculators](https:\/\/www.symbolab.com\/calculators) [Graphing Calculator](https:\/\/www.symbolab.com\/graphing-calculator) [Geometry Calculator](https:\/\/www.symbolab.com\/geometry-calculator) [Verify Solution](https:\/\/www.symbolab.com\/verify)\n\nApps [Symbolab App (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.devsense.symbolab) [Graphing Calculator (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.symbolab.graphingcalculator) [Practice (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.symbolab.practice) [Symbolab App (iOS)](https:\/\/apps.apple.com\/us\/app\/id876942533) [Graphing Calculator (iOS)](https:\/\/apps.apple.com\/us\/app\/id1435182798) [Practice (iOS)](https:\/\/apps.apple.com\/us\/app\/id1469186281) [Chrome Extension](https:\/\/chromewebstore.google.com\/detail\/mgfcnnpakbfcefgphceolkjldjfpieff)\n\nCompany [About Symbolab](https:\/\/www.symbolab.com\/about) [Blog](https:\/\/blog.symbolab.com\/) [Help](https:\/\/www.symbolab.com\/help) [Contact Us](https:\/\/www.symbolab.com\/solver\/integral-calculator)\n\nLegal [Privacy](https:\/\/www.symbolab.com\/privacy-policy) [Service Terms](https:\/\/www.symbolab.com\/terms-of-use) [Cookie Policy](https:\/\/www.symbolab.com\/cookie-policy) Cookie Settings Copyright, Community Guidelines, DSA & other Legal Resources [Learneo Legal Center](https:\/\/learneo.com\/legal)\n\n[Feedback](https:\/\/www.symbolab.com\/solver\/integral-calculator)\n\nSocial Media\n\nSymbolab, a Learneo, Inc. business\n\n© Learneo, Inc. 2024\n***\n(optional)\n\n(optional)\n\nPlease add a message.\n\nMessage received. Thanks for the feedback.\n\nCancel **Send**\n\n![](https:\/\/www.symbolab.com\/public\/img\/ajax-loader.gif)\n\n![]()","attrs_readable_markdown":"## Introduction to Integration\n### What is an Integration?\nIntegration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the [graph of a function](https:\/\/www.symbolab.com\/graphing-calculator). Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form.\n\nWe distinguish integration into two forms: [definite](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator) and [indefinite integrals](https:\/\/www.symbolab.com\/solver\/indefinite-integral-calculator). Fundamental instruments in [calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator), differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches.\n\n### Integration - An Inverse Process of Differentiation\nIntegration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one.\n\nIf \\$\\\\frac{d(F(x))}{dx}=f(x)\\$, then \\$\\\\int f(x)dx=F(x)+C\\$ . This is known as indefinite integrals.\n\n**For Example**\n\nSuppose \\$f(x)=x^3\\$\n\nThe [derivative](https:\/\/www.symbolab.com\/solver\/derivative-calculator) of f(x) is \\$f'(x)=3x^2\\$\n\nThe [antiderivative](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator) of \\$3x^2\\$ is \\$x^3\\$\n\nSo the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is \\$ \\\\int 3x^2dx=x^3+C\\$\n\nTherefore, if \\$\\\\frac{dy}{dx}=f(x)\\$, then we can write it as y = \\$\\\\int f(x)dx=F(x)+C\\$ where:\n\n- \\$\\\\int f(x)dx\\$ will represent the complete class of integral.\n- C is an arbitrary constant.\n- x is the variable of equation.\n- The symbol \\$\\\\int\\$ denotes the integral.\n- f(x) is the integrand.\n### Rules of Integration\n**Sum and Difference Rules:**\n\n\\$\\\\int \\[f(x) + g(x)\\] dx = \\\\int f(x) dx + \\\\int g(x) dx\\$\n\n\\$\\\\int \\[f(x) - g(x)\\] dx = \\\\int f(x) dx - \\\\int g(x) dx\\$\n\nFor example: \\$\\\\int (x^2 + 3x) dx = \\\\int x^2 dx + \\\\int 3x dx\\$\n\n\\$= \\\\frac{x^3}{3} + \\\\frac{3x^2}{2} + C\\$\n\n**Power Rule:**\n\n\\$ \\\\int x^ndx= \\\\frac {x^{(n+1)}}{n+1} +C \\$\n\nPlease note here n\\$\\\\neq\\$-1\n\nFor example: \\$\\\\int x^5dx=\\\\frac{x^6}{6}+C\\$\n\n**Exponential Rules:**\n\n\\$ \\\\int e^xdx=e^x+C\\$\n\n\\$ \\\\int a^xdx= \\\\frac{a^x}{ln(a)}+C\\$\n\n\\$ \\\\int ln(x)dx=xln(x)-x+C\\$\n\n**Constant Multiplication Rule:**\n\n\\$ \\\\int adx=ax+C \\$\n\n**Reciprocal Rule:**\n\n\\$ \\\\int \\\\frac{1}{x}dx=ln\\|x\\|+C \\$\n\n### Properties of Integration\nProperties of indefinite\n\n\\$\\\\int \\[f(x) \\\\pm g(x)\\]dx= \\\\int f(x)dx \\\\pm \\\\int g(x)dx \\$\n\n\\$ \\\\int kf(x)dx = k \\\\int f(x)dx \\$ (here k is the constant)\n\n\\$ \\\\int f(x)dx= \\\\int g(x)dx \\$ if \\$ \\\\int \\[f(x)-g(x)\\]dx=0 \\$\n\nBy collabrating these properties, we derive: \\$\\\\int \\\\sum k \\_nf\\_n(x)dx=\\\\sum k\\_n\\\\int f\\_n(x)dx \\$\n\n### What are some uses for an integral calculator and what is it exactly?\nOne may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals.\n\n### Implementations of an integral calculator:\n1. Two examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative.\n2. Double-checking hand computations helps one verify integration answers.\n3. \"Handling complex functions\" is the capacity to combine activities that are challenging for manual handling.\n4. Applications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs.\n5. Improving learning means giving students help understanding integration techniques and their uses.\n\nFor example using an integration calculator we can find:\n\n\\$\\\\int (3x^2-2x+1)dx =3\\\\frac{x^{2+1}}{2+1}-2 \\\\frac{x^{1+1}}{1+1}+x+C\\$ \\$=3\\\\frac{x^{3}}{3}-2 \\\\frac{x^{2}}{2}+x+C\\$ \\$=x^3-x^2+x+C\\$\n\n### Methods of Integration\nThe simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here:\n\n**1\\. Integration by Decomposition**\n\nIn this method we need to breakthe function into basic parts: \\$\\$\\\\int \\\\frac{x^2-x+1}{x^3}dx\\$\\$ Expanding: \\$\\$\\\\int (\\\\frac{x^2}{x^3}-\\\\frac{x}{x^3}+\\\\frac{1}{x^3})dx \\$\\$ Applying the basic rules: \\$ log \\|x\\|+ \\\\frac {1}{x}-\\\\frac{1}{2x^2}+C\\$\n\n**2\\. Integration by Substitution**\n\nFor simplyfing the integral change the variables: \\$\\$ \\\\int sin(mx)dx \\$\\$ Let mx=t, so \\$dx=dt\/m\\$ Therefore, \\$ \\\\frac{1}{m} \\\\int sin t dt= -\\\\frac{1}{m}cos t +C \\$\n\n**3\\. Integration using Partial Fractions**\n\nWe use this function for rational functions: \\$ \\\\int \\\\frac{1}{(x+1)(x+2)}dx=\\\\frac{A}{x+1}+ \\\\frac{B}{x+2} \\$ so now we can integrate them seperately and solve for the value of A and B.\n\n**4\\. Integration by Parts**\n\nDerived from the product rule of differentiation: \\$\\$ \\\\int udv = uv - \\\\int vdu \\$\\$\n\n### Applications of Integration\nIntegration has great use in domains like physics, engineering, and economics. Of the greatest significance are:\n\n1. One may get areas by use of the area under curves.\n2. The volume computation is mostly about figuring the solid of rotation volume.\n3. Applications of physics abound in motion problems, work done, and the center of mass.\n4. Calculation of probability density functions makes use of statistics and probability.\n\nThis extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus.\n\nFrequently Asked Questions (FAQ)\n\n- What is the use of integration in real life?\n- Integrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models.\n- What is the best integral calculator?\n- Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more.\n- What does to integrate mean?\n- Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.","meta_canonical":null}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

6 days ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0.2 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://www.symbolab.com/solver/integral-calculator
Last Crawled	2026-04-05 15:26:52 (6 days ago)
First Indexed	2015-01-16 22:19:58 (11 years ago)
HTTP Status Code	200
Meta Title	Integral Calculator
Meta Description	Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Type in any integral to get the solution, steps and graph
Meta Canonical	null
Boilerpipe Text	Introduction to Integration What is an Integration? Integration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the graph of a function . Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form. We distinguish integration into two forms: definite and indefinite integrals . Fundamental instruments in calculus , differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches. Integration - An Inverse Process of Differentiation Integration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one. If $\frac{d(F(x))}{dx}=f(x)$, then $\int f(x)dx=F(x)+C$ . This is known as indefinite integrals. For Example Suppose $f(x)=x^3$ The derivative of f(x) is $f'(x)=3x^2$ The antiderivative of $3x^2$ is $x^3$ So the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is $ \int 3x^2dx=x^3+C$ Therefore, if $\frac{dy}{dx}=f(x)$, then we can write it as y = $\int f(x)dx=F(x)+C$ where: $\int f(x)dx$ will represent the complete class of integral. C is an arbitrary constant. x is the variable of equation. The symbol $\int$ denotes the integral. f(x) is the integrand. Rules of Integration Sum and Difference Rules: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$ $\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx$ For example: $\int (x^2 + 3x) dx = \int x^2 dx + \int 3x dx$ $= \frac{x^3}{3} + \frac{3x^2}{2} + C$ Power Rule: $ \int x^ndx= \frac {x^{(n+1)}}{n+1} +C $ Please note here n$\neq$-1 For example: $\int x^5dx=\frac{x^6}{6}+C$ Exponential Rules: $ \int e^xdx=e^x+C$ $ \int a^xdx= \frac{a^x}{ln(a)}+C$ $ \int ln(x)dx=xln(x)-x+C$ Constant Multiplication Rule: $ \int adx=ax+C $ Reciprocal Rule: $ \int \frac{1}{x}dx=ln\|x\|+C $ Properties of Integration Properties of indefinite $\int [f(x) \pm g(x)]dx= \int f(x)dx \pm \int g(x)dx $ $ \int kf(x)dx = k \int f(x)dx $ (here k is the constant) $ \int f(x)dx= \int g(x)dx $ if $ \int [f(x)-g(x)]dx=0 $ By collabrating these properties, we derive: $\int \sum k _nf_n(x)dx=\sum k_n\int f_n(x)dx $ What are some uses for an integral calculator and what is it exactly? One may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals. Implementations of an integral calculator: Two examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative. Double-checking hand computations helps one verify integration answers. "Handling complex functions" is the capacity to combine activities that are challenging for manual handling. Applications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs. Improving learning means giving students help understanding integration techniques and their uses. For example using an integration calculator we can find: $\int (3x^2-2x+1)dx =3\frac{x^{2+1}}{2+1}-2 \frac{x^{1+1}}{1+1}+x+C$ $=3\frac{x^{3}}{3}-2 \frac{x^{2}}{2}+x+C$ $=x^3-x^2+x+C$ Methods of Integration The simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here: 1. Integration by Decomposition In this method we need to breakthe function into basic parts: $$\int \frac{x^2-x+1}{x^3}dx$$ Expanding: $$\int (\frac{x^2}{x^3}-\frac{x}{x^3}+\frac{1}{x^3})dx $$ Applying the basic rules: $ log \|x\|+ \frac {1}{x}-\frac{1}{2x^2}+C$ 2. Integration by Substitution For simplyfing the integral change the variables: $$ \int sin(mx)dx $$ Let mx=t, so $dx=dt/m$ Therefore, $ \frac{1}{m} \int sin t dt= -\frac{1}{m}cos t +C $ 3. Integration using Partial Fractions We use this function for rational functions: $ \int \frac{1}{(x+1)(x+2)}dx=\frac{A}{x+1}+ \frac{B}{x+2} $ so now we can integrate them seperately and solve for the value of A and B. 4. Integration by Parts Derived from the product rule of differentiation: $$ \int udv = uv - \int vdu $$ Applications of Integration Integration has great use in domains like physics, engineering, and economics. Of the greatest significance are: One may get areas by use of the area under curves. The volume computation is mostly about figuring the solid of rotation volume. Applications of physics abound in motion problems, work done, and the center of mass. Calculation of probability density functions makes use of statistics and probability. This extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus. Frequently Asked Questions (FAQ) What is the use of integration in real life? Integrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models. What is the best integral calculator? Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. What does to integrate mean? Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.
Markdown	You can see your coupon in the [user page](https://www.symbolab.com/user#subscription) [Go To QuillBot Upgrade to Pro Continue to site]() [We've updated our]() [Privacy Policy](https://www.symbolab.com/privacy-policy) effective December 15. Please read our updated Privacy Policy and tap Continue - [Solutions](https://www.symbolab.com/solver) [Integral Calculator](https://www.symbolab.com/solver/integral-calculator) [Derivative Calculator](https://www.symbolab.com/solver/derivative-calculator) [Algebra Calculator](https://www.symbolab.com/solver/algebra-calculator) [Matrix Calculator](https://www.symbolab.com/solver/matrix-calculator) [More...](https://www.symbolab.com/solver) - [Graphing](https://www.symbolab.com/graphing-calculator) [Line Graph Calculator](https://www.symbolab.com/graphing-calculator/line) [Exponential Graph Calculator](https://www.symbolab.com/graphing-calculator/exponential-graph) [Quadratic Graph Calculator](https://www.symbolab.com/graphing-calculator/quadratic-graph) [Sine Graph Calculator](https://www.symbolab.com/graphing-calculator/sine-graph) [More...](https://www.symbolab.com/graphing-calculator#examples=true) - [Calculators](https://www.symbolab.com/calculators) [BMI Calculator](https://www.symbolab.com/calculator/fitness/bmi) [Compound Interest Calculator](https://www.symbolab.com/calculator/finance/compound_interest_rate) [Percentage Calculator](https://www.symbolab.com/calculator/math/percentage) [Acceleration Calculator](https://www.symbolab.com/calculator/physics/acceleration) [More...](https://www.symbolab.com/calculators) - [Geometry](https://www.symbolab.com/geometry-calculator) [Pythagorean Theorem Calculator](https://www.symbolab.com/geometry-calculator/pythagorean-theorem-calculator) [Circle Area Calculator](https://www.symbolab.com/geometry-calculator/circle-area-diameter-calculator) [Isosceles Triangle Calculator](https://www.symbolab.com/geometry-calculator/isosceles-triangle-prove-angles-calculator) [Triangles Calculator](https://www.symbolab.com/geometry-calculator/triangle-perimeter-calculator) [More...](https://www.symbolab.com/geometry-calculator#examples=true) - [AI Chat](https://www.symbolab.com/ai-chat) - [Tools](https://www.symbolab.com/solver/integral-calculator) [Notebook](https://www.symbolab.com/notebook) [Groups](https://www.symbolab.com/groups) [Cheat Sheets](https://www.symbolab.com/cheat-sheets) [Worksheets](https://www.symbolab.com/worksheets/Pre-Algebra) [Study Guides](https://www.symbolab.com/study-guides) [Practice](https://www.symbolab.com/practice) [Verify Solution](https://www.symbolab.com/verify) - [Upgrade](https://www.symbolab.com/solver/integral-calculator) Close Accept Terms Good job\! ![Practice](https://www.symbolab.com/public/images/practice_icon1.png) Practice More Type your Answer Verify \| \| \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| x^2 \| x^{\\msquare} \| \\log\_{\\msquare} \| \\sqrt{\\square} \| \\nthroot\[\\msquare\]{\\square} \| \\le \| \\ge \| \\frac{\\msquare}{\\msquare} \| \\cdot \| \\div \| x^{\\circ} \| \\pi \| \| \\left(\\square\\right)^{'} \| \\frac{d}{dx} \| \\frac{\\partial}{\\partial x} \| \\int \| \\int\_{\\msquare}^{\\msquare} \| \\lim \| \\sum \| \\infty \| \\theta \| (f\\:\\circ\\:g) \| f(x) \| ![](https://www.symbolab.com/public/img/pad_matrixnxn.png) \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\| \| ▭\\:\\longdivision{▭} \| \\times \\twostack{▭}{▭} \| \+ \\twostack{▭}{▭} \| \- \\twostack{▭}{▭} \| \\left( \| \\right) \| \\times \| \\square\\frac{\\square}{\\square} \| Take a challenge Subscribe to verify your answer Subscribe Are you sure you want to leave this Challenge? By closing this window you will lose this challenge Cancel Leave - [Pre Algebra](https://www.symbolab.com/solver/pre-algebra-calculator) [Order of Operations](https://www.symbolab.com/solver/order-of-operations-calculator) [Factors & Primes](https://www.symbolab.com/solver/factors-primes-calculator) [Fractions](https://www.symbolab.com/solver/fractions-calculator) [Long Arithmetic](https://www.symbolab.com/solver/long-arithmetic-calculator) [Decimals](https://www.symbolab.com/solver/decimals-calculator) [Exponents & Radicals](https://www.symbolab.com/solver/exponents-radicals-calculator) [Ratios & Proportions](https://www.symbolab.com/solver/ratios-proportions-calculator) [Percent](https://www.symbolab.com/solver/percent-calculator) [Modulo](https://www.symbolab.com/solver/modulo-calculator) [Number Line](https://www.symbolab.com/solver/number-line-calculator) [Expanded Form](https://www.symbolab.com/solver/expanded-form-calculator) [Mean, Median & Mode](https://www.symbolab.com/solver/mean-median-mode-calculator) - [Algebra](https://www.symbolab.com/solver/algebra-calculator) [Equations](https://www.symbolab.com/solver/equation-calculator) [Inequalities](https://www.symbolab.com/solver/inequalities-calculator) [System of Equations](https://www.symbolab.com/solver/system-of-equations-calculator) [System of Inequalities](https://www.symbolab.com/solver/system-of-inequalities-calculator) [Testing Solutions](https://www.symbolab.com/solver/testing-solutions-calculator) [Basic Operations](https://www.symbolab.com/solver/basic-operations-calculator) [Algebraic Properties](https://www.symbolab.com/solver/algebraic-properties-calculator) [Partial Fractions](https://www.symbolab.com/solver/partial-fractions-calculator) [Polynomials](https://www.symbolab.com/solver/polynomial-calculator) [Rational Expressions](https://www.symbolab.com/solver/rational-expression-calculator) [Sequences](https://www.symbolab.com/solver/sequence-calculator) [Power Sums](https://www.symbolab.com/solver/power-sums-calculator) [Interval Notation](https://www.symbolab.com/solver/interval-notation-calculator) [Pi (Product) Notation](https://www.symbolab.com/solver/pi-product-notation-calculator) [Induction](https://www.symbolab.com/solver/induction-calculator) [Prove That](https://www.symbolab.com/solver/prove-that-calculator) [Logical Sets](https://www.symbolab.com/solver/logical-sets-calculator) [Word Problems](https://www.symbolab.com/solver/word-problems-calculator) - [Pre Calculus](https://www.symbolab.com/solver/pre-calculus-calculator) [Equations](https://www.symbolab.com/solver/pre-calculus-equation-calculator) [Inequalities](https://www.symbolab.com/solver/pre-calculus-inequalities-calculator) [Scientific Calculator](https://www.symbolab.com/solver/scientific-calculator) [Scientific Notation Arithmetics](https://www.symbolab.com/solver/scientific-notation-arithmetics) [Complex Numbers](https://www.symbolab.com/solver/pre-calculus-complex-numbers-calculator) [Polar/Cartesian](https://www.symbolab.com/solver/polar-cartesian-calculator) [Simultaneous Equations](https://www.symbolab.com/solver/simultaneous-equations-calculator) [System of Inequalities](https://www.symbolab.com/solver/pre-calculus-system-of-inequalities-calculator) [Polynomials](https://www.symbolab.com/solver/pre-calculus-polynomial-calculator) [Rationales](https://www.symbolab.com/solver/rationales-calculator) [Functions](https://www.symbolab.com/solver/pre-calculus-functions-calculator) [Arithmetic & Comp.](https://www.symbolab.com/solver/pre-calculus-function-arithmetic-composition-calculator) [Coordinate Geometry](https://www.symbolab.com/solver/pre-calculus-coordinate-geometry-calculator) [Plane Geometry](https://www.symbolab.com/solver/plane-geometry-calculator) [Solid Geometry](https://www.symbolab.com/solver/solid-geometry-calculator) [Trigonometry](https://www.symbolab.com/solver/pre-calculus-trigonometry-calculator) - [Calculus](https://www.symbolab.com/solver/calculus-calculator) [Derivatives](https://www.symbolab.com/solver/derivative-calculator) [Derivative Applications](https://www.symbolab.com/solver/derivative-applications-calculator) [Limits](https://www.symbolab.com/solver/limit-calculator) [Integrals](https://www.symbolab.com/solver/integral-calculator) [Integral Applications](https://www.symbolab.com/solver/integral-applications-calculator) [Integral Approximation](https://www.symbolab.com/solver/integral-approximation-calculator) [Series](https://www.symbolab.com/solver/series-calculator) [ODE](https://www.symbolab.com/solver/ordinary-differential-equation-calculator) [Multivariable Calculus](https://www.symbolab.com/solver/multivariable-calculus-calculator) [Laplace Transform](https://www.symbolab.com/solver/laplace-calculator) [Taylor/Maclaurin Series](https://www.symbolab.com/solver/taylor-maclaurin-series-calculator) [Fourier Series](https://www.symbolab.com/solver/fourier-series-calculator) [Fourier Transform](https://www.symbolab.com/solver/fourier-transform-calculator) - [Functions](https://www.symbolab.com/solver/functions-line-calculator) [Line Equations](https://www.symbolab.com/solver/line-equation-calculator) [Functions](https://www.symbolab.com/solver/functions-calculator) [Arithmetic & Comp.](https://www.symbolab.com/solver/function-arithmetic-composition-calculator) [Conic Sections](https://www.symbolab.com/solver/conic-sections-calculator) [Transformation](https://www.symbolab.com/solver/function-transformation-calculator) - [Linear Algebra](https://www.symbolab.com/solver/linear-algebra-calculator) [Matrices](https://www.symbolab.com/solver/matrix-calculator) [Vectors](https://www.symbolab.com/solver/vector-calculator) - [Trigonometry](https://www.symbolab.com/solver/trigonometry-calculator) [Quadrant](https://www.symbolab.com/solver/quadrant-calculator) [Coterminal Angle](https://www.symbolab.com/solver/coterminal-angle-calculator) [Identities](https://www.symbolab.com/solver/trigonometric-identity-calculator) [Proving Identities](https://www.symbolab.com/solver/trigonometric-identity-proving-calculator) [Trig Equations](https://www.symbolab.com/solver/trigonometric-equation-calculator) [Trig Inequalities](https://www.symbolab.com/solver/trigonometric-inequality-calculator) [Evaluate Functions](https://www.symbolab.com/solver/trigonometric-function-calculator) [Simplify](https://www.symbolab.com/solver/trigonometric-simplification-calculator) - [Statistics](https://www.symbolab.com/solver/statistics-calculator) [Mean](https://www.symbolab.com/solver/arithmetic-mean-calculator) [Geometric Mean](https://www.symbolab.com/solver/geometric-mean-calculator) [Quadratic Mean](https://www.symbolab.com/solver/quadratic-mean-calculator) [Average](https://www.symbolab.com/solver/average-calculator) [Median](https://www.symbolab.com/solver/median-calculator) [Mode](https://www.symbolab.com/solver/mode-calculator) [Order](https://www.symbolab.com/solver/order-calculator) [Minimum](https://www.symbolab.com/solver/minimum-calculator) [Maximum](https://www.symbolab.com/solver/maximum-calculator) [Probability](https://www.symbolab.com/solver/probability-calculator) [Mid-Range](https://www.symbolab.com/solver/mid-range-calculator) [Range](https://www.symbolab.com/solver/range-calculator) [Standard Deviation](https://www.symbolab.com/solver/standard-deviation-calculator) [Variance](https://www.symbolab.com/solver/variance-calculator) [Lower Quartile](https://www.symbolab.com/solver/lower-quartile-calculator) [Upper Quartile](https://www.symbolab.com/solver/upper-quartile-calculator) [Interquartile Range](https://www.symbolab.com/solver/interquartile-range-calculator) [Midhinge](https://www.symbolab.com/solver/midhinge-calculator) [Standard Normal Distribution](https://www.symbolab.com/solver/standard-normal-distribution-calculator) - [Physics](https://www.symbolab.com/solver/physics-calculator) [Mechanics](https://www.symbolab.com/solver/mechanics-calculator) - [Chemistry](https://www.symbolab.com/solver/chemistry-calculator) [Chemical Reactions](https://www.symbolab.com/solver/chemical-reaction-calculator) [Chemical Properties](https://www.symbolab.com/solver/chemical-properties-calculator) - [Finance](https://www.symbolab.com/solver/finance-calculator) [Simple Interest](https://www.symbolab.com/solver/simple-interest-calculator) [Compound Interest](https://www.symbolab.com/solver/compound-interest-calculator) [Present Value](https://www.symbolab.com/solver/present-value-calculator) [Future Value](https://www.symbolab.com/solver/future-value-calculator) - [Economics](https://www.symbolab.com/solver/economics-calculator) [Point of Diminishing Return](https://www.symbolab.com/solver/point-of-diminishing-return-calculator) - [Conversions](https://www.symbolab.com/solver/conversion-calculator) [Currency](https://www.symbolab.com/solver/currency-converter) [Roman Numerals](https://www.symbolab.com/solver/roman-numeral-converter) [Radical to Exponent](https://www.symbolab.com/solver/radical-to-exponent-calculator) [Exponent to Radical](https://www.symbolab.com/solver/exponent-to-radical-calculator) [To Fraction](https://www.symbolab.com/solver/convert-to-fraction-calculator) [To Decimal](https://www.symbolab.com/solver/convert-to-decimal-calculator) [To Mixed Number](https://www.symbolab.com/solver/convert-to-mixed-number-calculator) [To Improper Fraction](https://www.symbolab.com/solver/convert-to-improper-fraction-calculator) [Radians to Degrees](https://www.symbolab.com/solver/radians-to-degrees-calculator) [Degrees to Radians](https://www.symbolab.com/solver/degrees-to-radians-calculator) [Degrees Minutes Seconds](https://www.symbolab.com/solver/degrees-minutes-seconds-calculator) [Hexadecimal](https://www.symbolab.com/solver/hexadecimal-converter) [Scientific Notation](https://www.symbolab.com/solver/scientific-notation-converter) [Distance](https://www.symbolab.com/solver/distance-converter) [Weight](https://www.symbolab.com/solver/weight-converter) [Time](https://www.symbolab.com/solver/time-converter) [Volume](https://www.symbolab.com/solver/volume-converter) ### Special Offer ## Double the Tools. One Smart Bundle. to master your math and writing\! Pro - Full access to solution steps - Full access to AI chat - Practice and improve - Access from any device - No ads - Unlimited storage - Detect AI-generated content - Advanced Grammar Checker - AI Humanizer - Paraphrase in unlimited modes - Summarize any text - Much More\! [Restrictions apply, check out our FAQs.](https://www.symbolab.com/bundle-faq) Go back Purchase Bundle ### Back to School Promotion Annual Annual - \$ % OFF Annual plan One time offer for one year, then \$ Go back \+ ## qb-banner-title ### ## ### [Solutions](https://www.symbolab.com/solver) \> [Calculus Calculator](https://www.symbolab.com/solver/calculus-calculator) \> # Integral Calculator Topic - [Pre Algebra](https://www.symbolab.com/solver/pre-algebra-calculator) - [Algebra](https://www.symbolab.com/solver/algebra-calculator) - [Pre Calculus](https://www.symbolab.com/solver/pre-calculus-calculator) - [Calculus](https://www.symbolab.com/solver/calculus-calculator) - [Derivatives]() - [First Derivative](https://www.symbolab.com/solver/first-derivative-calculator) - [WRT](https://www.symbolab.com/solver/derivative-with-respect-calculator) - [Specify Method]() - [Chain Rule](https://www.symbolab.com/solver/derivative-chain-rule-calculator) - [Product Rule](https://www.symbolab.com/solver/derivative-product-rule-calculator) - [Quotient Rule](https://www.symbolab.com/solver/derivative-quotient-rule-calculator) - [Sum/Diff Rule](https://www.symbolab.com/solver/derivative-sum-diff-rule-calculator) - [Second Derivative](https://www.symbolab.com/solver/second-derivative-calculator) - [Third Derivative](https://www.symbolab.com/solver/third-derivative-calculator) - [Higher Order Derivatives](https://www.symbolab.com/solver/higher-order-derivative-calculator) - [Derivative at a point](https://www.symbolab.com/solver/derivative-point-calculator) - [Partial Derivative](https://www.symbolab.com/solver/partial-derivative-calculator) - [Implicit Derivative](https://www.symbolab.com/solver/implicit-derivative-calculator) - [Second Implicit Derivative](https://www.symbolab.com/solver/second-implicit-derivative-calculator) - [Derivative using Definition](https://www.symbolab.com/solver/derivative-using-definition-calculator) - [Derivative Applications]() - [Tangent](https://www.symbolab.com/solver/tangent-line-calculator) - [Slope of Tangent](https://www.symbolab.com/solver/slope-of-tangent-calculator) - [Normal](https://www.symbolab.com/solver/normal-line-calculator) - [Curved Line Slope](https://www.symbolab.com/solver/curved-line-slope-calculator) - [Extreme Points](https://www.symbolab.com/solver/calculus-function-extreme-points-calculator) - [Tangent to Conic](https://www.symbolab.com/solver/tangent-to-conic-calculator) - [Linear Approximation](https://www.symbolab.com/solver/linear-approximation-calculator) - [Difference Quotient](https://www.symbolab.com/solver/difference-quotient-calculator) - [Horizontal Tangent](https://www.symbolab.com/solver/horizontal-tangent-calculator) - [Limits]() - [One Variable](https://www.symbolab.com/solver/single-var-limit-calculator) - [Multi Variable Limit](https://www.symbolab.com/solver/multi-var-limit-calculator) - [One Sided](https://www.symbolab.com/solver/one-sided-limit-calculator) - [At Infinity](https://www.symbolab.com/solver/limit-infinity-calculator) - [Specify Method]() - [L'Hopital's Rule](https://www.symbolab.com/solver/limit-lhopital-calculator) - [Squeeze Theorem](https://www.symbolab.com/solver/limit-squeeze-theorem-calculator) - [Chain Rule](https://www.symbolab.com/solver/limit-chain-rule-calculator) - [Factoring](https://www.symbolab.com/solver/limit-using-factoring-calculator) - [Substitution](https://www.symbolab.com/solver/limit-substitution-calculator) - [Sandwich Theorem](https://www.symbolab.com/solver/limit-sandwich-theorem-calculator) - [Integrals]() - [Indefinite Integrals](https://www.symbolab.com/solver/indefinite-integral-calculator) - [Definite Integrals](https://www.symbolab.com/solver/definite-integral-calculator) - [Specific-Method]() - [Partial Fractions](https://www.symbolab.com/solver/partial-fractions-integration-calculator) - [U-Substitution](https://www.symbolab.com/solver/u-substitution-integration-calculator) - [Trigonometric Substitution](https://www.symbolab.com/solver/trigonometric-substitution-integration-calculator) - [Weierstrass Substitution](https://www.symbolab.com/solver/weierstrass-substitution-calculator) - [By Parts](https://www.symbolab.com/solver/by-parts-integration-calculator) - [Long Division](https://www.symbolab.com/solver/long-division-integration-calculator) - [Improper Integrals](https://www.symbolab.com/solver/improper-integral-calculator) - [Antiderivatives](https://www.symbolab.com/solver/antiderivative-calculator) - [Double Integrals](https://www.symbolab.com/solver/double-integrals-calculator) - [Triple Integrals](https://www.symbolab.com/solver/triple-integrals-calculator) - [Multiple Integrals](https://www.symbolab.com/solver/multiple-integrals-calculator) - [Integral Applications]() - [Limit of Sum](https://www.symbolab.com/solver/limit-of-sum-calculator) - [Area under curve](https://www.symbolab.com/solver/area-under-curve-calculator) - [Area between curves](https://www.symbolab.com/solver/area-between-curves-calculator) - [Area under polar curve](https://www.symbolab.com/solver/area-under-polar-curve-calculator) - [Volume of solid of revolution](https://www.symbolab.com/solver/volume-calculator) - [Arc Length](https://www.symbolab.com/solver/arc-length-calculator) - [Function Average](https://www.symbolab.com/solver/function-average-calculator) - [Integral Approximation]() - [Riemann Sum](https://www.symbolab.com/solver/riemann-sum-calculator) - [Trapezoidal](https://www.symbolab.com/solver/trapezoidal-approximation-calculator) - [Simpson's Rule](https://www.symbolab.com/solver/simpsons-rule-calculator) - [Midpoint Rule](https://www.symbolab.com/solver/midpoint-rule-calculator) - [Series]() - [Convergence]() - [Geometric Series Test](https://www.symbolab.com/solver/geometric-series-test-calculator) - [Telescoping Series Test](https://www.symbolab.com/solver/telescoping-series-test-calculator) - [Alternating Series Test](https://www.symbolab.com/solver/alternating-series-test-calculator) - [P Series Test](https://www.symbolab.com/solver/p-series-test-calculator) - [Divergence Test](https://www.symbolab.com/solver/series-divergence-test-calculator) - [Ratio Test](https://www.symbolab.com/solver/series-ratio-test-calculator) - [Root Test](https://www.symbolab.com/solver/series-root-test-calculator) - [Comparison Test](https://www.symbolab.com/solver/series-comparison-test-calculator) - [Limit Comparison Test](https://www.symbolab.com/solver/series-limit-comparison-test-calculator) - [Integral Test](https://www.symbolab.com/solver/series-integral-test-calculator) - [Absolute Convergence](https://www.symbolab.com/solver/series-absolute-convergence-calculator) - [Power Series]() - [Radius of Convergence](https://www.symbolab.com/solver/radius-of-convergence-calculator) - [Interval of Convergence](https://www.symbolab.com/solver/interval-of-convergence-calculator) - [ODE]() - [Linear First Order](https://www.symbolab.com/solver/linear-first-order-differential-equation-calculator) - [Linear w/constant coefficients](https://www.symbolab.com/solver/linear-constant-coefficients-differential-equation-calculator) - [Separable](https://www.symbolab.com/solver/separable-differential-equation-calculator) - [Bernoulli](https://www.symbolab.com/solver/bernoulli-differential-equation-calculator) - [Exact](https://www.symbolab.com/solver/exact-differential-equation-calculator) - [Second Order](https://www.symbolab.com/solver/second-order-differential-equation-calculator) - [Homogenous](https://www.symbolab.com/solver/homogenous-differential-equation-calculator) - [Non Homogenous](https://www.symbolab.com/solver/non-homogenous-differential-equation-calculator) - [Substitution](https://www.symbolab.com/solver/substitution-differential-equation-calculator) - [System of ODEs](https://www.symbolab.com/solver/system-of-differential-equations-calculator) - [IVP using Laplace](https://www.symbolab.com/solver/ivp-laplace-calculator) - [Series Solutions](https://www.symbolab.com/solver/ode-series-solutions-calculator) - [Method of Frobenius](https://www.symbolab.com/solver/frobenius-method-calculator) - [Gamma Function](https://www.symbolab.com/solver/gamma-function-calculator) - [Multivariable Calculus]() - [Partial Derivative](https://www.symbolab.com/solver/multivariable-partial-derivative-calculator) - [Implicit Derivative](https://www.symbolab.com/solver/multivariable-implicit-derivative-calculator) - [Tangent to Conic](https://www.symbolab.com/solver/multivariable-tangent-to-conic-calculator) - [Multi Variable Limit](https://www.symbolab.com/solver/multi-var-calculus-limit-calculator) - [Multiple Integrals](https://www.symbolab.com/solver/multiple-integrals-multivariable-calculator) - [Gradient](https://www.symbolab.com/solver/gradient-calculator) - [Divergence](https://www.symbolab.com/solver/divergence-calculator) - [Extreme Points](https://www.symbolab.com/solver/multi-var-function-extreme-points-calculator) - [Laplace Transform]() - [Inverse](https://www.symbolab.com/solver/inverse-laplace-calculator) - [Taylor/Maclaurin Series]() - [Taylor Series](https://www.symbolab.com/solver/taylor-series-calculator) - [Maclaurin Series](https://www.symbolab.com/solver/maclaurin-series-calculator) - [Fourier Series](https://www.symbolab.com/solver/fourier-series-calculator) - [Fourier Transform](https://www.symbolab.com/solver/fourier-transform-calculator) - [Functions](https://www.symbolab.com/solver/functions-line-calculator) - [Linear Algebra](https://www.symbolab.com/solver/linear-algebra-calculator) - [Trigonometry](https://www.symbolab.com/solver/trigonometry-calculator) - [Statistics](https://www.symbolab.com/solver/statistics-calculator) - [Physics](https://www.symbolab.com/solver/physics-calculator) - [Chemistry](https://www.symbolab.com/solver/chemistry-calculator) - [Finance](https://www.symbolab.com/solver/finance-calculator) - [Economics](https://www.symbolab.com/solver/economics-calculator) - [Conversions](https://www.symbolab.com/solver/conversion-calculator) Add to Chrome Get our extension, you can capture any math problem from any website Full pad \| \| \| \| \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| x^2 \| x^{\\msquare} \| \\log\_{\\msquare} \| \\sqrt{\\square} \| \\nthroot\[\\msquare\]{\\square} \| \\le \| \\ge \| \\frac{\\msquare}{\\msquare} \| \\cdot \| \\div \| x^{\\circ} \| \\pi \| \| \\left(\\square\\right)^{'} \| \\frac{d}{dx} \| \\frac{\\partial}{\\partial x} \| \\int \| \\int\_{\\msquare}^{\\msquare} \| \\lim \| \\sum \| \\infty \| \\theta \| (f\\:\\circ\\:g) \| f(x) \| ![](https://www.symbolab.com/public/img/pad_matrixnxn.png) \| \| \| \| \| \| \| \| \| \|---\|---\|---\|---\|---\|---\|---\| \| \- \\twostack{▭}{▭} \| \\lt \| 7 \| 8 \| 9 \| \\div \| AC \| \| \+ \\twostack{▭}{▭} \| \\gt \| 4 \| 5 \| 6 \| \\times \| \\square\\frac{\\square}{\\square} \| \| \\times \\twostack{▭}{▭} \| \\left( \| 1 \| 2 \| 3 \| \- \| x \| \| ▭\\:\\longdivision{▭} \| \\right) \| . \| 0 \| \= \| \+ \| y \| \\mathrm{partial\\:fractions} \\mathrm{substitution} \\mathrm{long\\:division} \\mathrm{trigonometric\\:substitution} \\mathrm{by\\:parts} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test Go [Steps](https://www.symbolab.com/solver/integral-calculator#steps) [Graph](https://www.symbolab.com/solver/integral-calculator#plot) [Related](https://www.symbolab.com/solver/integral-calculator#related_anchor) [Examples](https://www.symbolab.com/solver/integral-calculator#Examples) Generated by AI AI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view. Verify your Answer Subscribe to verify your answer Subscribe Save to Notebook\! Sign in to save notes Sign in Verify Save Show Steps Hide Steps ## Number Line ![]() ![](https://www.symbolab.com/public/img/ajax-loader.gif) Related Integral Examples - [\\int e^x\\cos(x)dx](https://www.symbolab.com/solver/integral-calculator/%5Cint%20e%5Ex%5Ccos%28x%29dx?or=ex) - [\\int \\cos^3(x)\\sin (x)dx](https://www.symbolab.com/solver/integral-calculator/%5Cint%20%5Ccos%5E3%28x%29%5Csin%20%28x%29dx?or=ex) - [\\int \\frac{2x+1}{(x+5)^3}](https://www.symbolab.com/solver/integral-calculator/%5Cint%20%5Cfrac%7B2x%2B1%7D%7B%28x%2B5%29%5E3%7D?or=ex) - [\\int\_{0}^{\\pi}\\sin(x)dx](https://www.symbolab.com/solver/integral-calculator/%5Cint_%7B0%7D%5E%7B%5Cpi%7D%5Csin%28x%29dx?or=ex) - [\\int\_{a}^{b} x^2dx](https://www.symbolab.com/solver/integral-calculator/%5Cint_%7Ba%7D%5E%7Bb%7D%20x%5E2dx?or=ex) - [\\int\_{0}^{2\\pi}\\cos^2(\\theta)d\\theta](https://www.symbolab.com/solver/integral-calculator/%5Cint_%7B0%7D%5E%7B2%5Cpi%7D%5Ccos%5E2%28%5Ctheta%29d%5Ctheta?or=ex) - [partial\\:fractions\\:\\int\_{0}^{1} \\frac{32}{x^{2}-64}dx](https://www.symbolab.com/solver/integral-calculator/partial%20fractions%20%5Cint_%7B0%7D%5E%7B1%7D%20%5Cfrac%7B32%7D%7Bx%5E%7B2%7D-64%7Ddx?or=ex) - [substitution\\:\\int\\frac{e^{x}}{e^{x}+e^{-x}}dx,\\:u=e^{x}](https://www.symbolab.com/solver/integral-calculator/substitution%20%5Cint%5Cfrac%7Be%5E%7Bx%7D%7D%7Be%5E%7Bx%7D%2Be%5E%7B-x%7D%7Ddx%2C%20u%3De%5E%7Bx%7D?or=ex) - Show More ## Introduction to Integration ### What is an Integration? Integration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the [graph of a function](https://www.symbolab.com/graphing-calculator). Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form. We distinguish integration into two forms: [definite](https://www.symbolab.com/solver/definite-integral-calculator) and [indefinite integrals](https://www.symbolab.com/solver/indefinite-integral-calculator). Fundamental instruments in [calculus](https://www.symbolab.com/solver/calculus-calculator), differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches. ### Integration - An Inverse Process of Differentiation Integration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one. If \$\\frac{d(F(x))}{dx}=f(x)\$, then \$\\int f(x)dx=F(x)+C\$ . This is known as indefinite integrals. For Example Suppose \$f(x)=x^3\$ The [derivative](https://www.symbolab.com/solver/derivative-calculator) of f(x) is \$f'(x)=3x^2\$ The [antiderivative](https://www.symbolab.com/solver/antiderivative-calculator) of \$3x^2\$ is \$x^3\$ So the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is \$ \\int 3x^2dx=x^3+C\$ Therefore, if \$\\frac{dy}{dx}=f(x)\$, then we can write it as y = \$\\int f(x)dx=F(x)+C\$ where: - \$\\int f(x)dx\$ will represent the complete class of integral. - C is an arbitrary constant. - x is the variable of equation. - The symbol \$\\int\$ denotes the integral. - f(x) is the integrand. ### Rules of Integration Sum and Difference Rules: \$\\int \[f(x) + g(x)\] dx = \\int f(x) dx + \\int g(x) dx\$ \$\\int \[f(x) - g(x)\] dx = \\int f(x) dx - \\int g(x) dx\$ For example: \$\\int (x^2 + 3x) dx = \\int x^2 dx + \\int 3x dx\$ \$= \\frac{x^3}{3} + \\frac{3x^2}{2} + C\$ Power Rule: \$ \\int x^ndx= \\frac {x^{(n+1)}}{n+1} +C \$ Please note here n\$\\neq\$-1 For example: \$\\int x^5dx=\\frac{x^6}{6}+C\$ Exponential Rules: \$ \\int e^xdx=e^x+C\$ \$ \\int a^xdx= \\frac{a^x}{ln(a)}+C\$ \$ \\int ln(x)dx=xln(x)-x+C\$ Constant Multiplication Rule: \$ \\int adx=ax+C \$ Reciprocal Rule: \$ \\int \\frac{1}{x}dx=ln\\|x\\|+C \$ ### Properties of Integration Properties of indefinite \$\\int \[f(x) \\pm g(x)\]dx= \\int f(x)dx \\pm \\int g(x)dx \$ \$ \\int kf(x)dx = k \\int f(x)dx \$ (here k is the constant) \$ \\int f(x)dx= \\int g(x)dx \$ if \$ \\int \[f(x)-g(x)\]dx=0 \$ By collabrating these properties, we derive: \$\\int \\sum k \_nf\_n(x)dx=\\sum k\_n\\int f\_n(x)dx \$ ### What are some uses for an integral calculator and what is it exactly? One may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals. ### Implementations of an integral calculator: 1. Two examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative. 2. Double-checking hand computations helps one verify integration answers. 3. "Handling complex functions" is the capacity to combine activities that are challenging for manual handling. 4. Applications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs. 5. Improving learning means giving students help understanding integration techniques and their uses. For example using an integration calculator we can find: \$\\int (3x^2-2x+1)dx =3\\frac{x^{2+1}}{2+1}-2 \\frac{x^{1+1}}{1+1}+x+C\$ \$=3\\frac{x^{3}}{3}-2 \\frac{x^{2}}{2}+x+C\$ \$=x^3-x^2+x+C\$ ### Methods of Integration The simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here: 1\. Integration by Decomposition In this method we need to breakthe function into basic parts: \$\$\\int \\frac{x^2-x+1}{x^3}dx\$\$ Expanding: \$\$\\int (\\frac{x^2}{x^3}-\\frac{x}{x^3}+\\frac{1}{x^3})dx \$\$ Applying the basic rules: \$ log \\|x\\|+ \\frac {1}{x}-\\frac{1}{2x^2}+C\$ 2\. Integration by Substitution For simplyfing the integral change the variables: \$\$ \\int sin(mx)dx \$\$ Let mx=t, so \$dx=dt/m\$ Therefore, \$ \\frac{1}{m} \\int sin t dt= -\\frac{1}{m}cos t +C \$ 3\. Integration using Partial Fractions We use this function for rational functions: \$ \\int \\frac{1}{(x+1)(x+2)}dx=\\frac{A}{x+1}+ \\frac{B}{x+2} \$ so now we can integrate them seperately and solve for the value of A and B. 4\. Integration by Parts Derived from the product rule of differentiation: \$\$ \\int udv = uv - \\int vdu \$\$ ### Applications of Integration Integration has great use in domains like physics, engineering, and economics. Of the greatest significance are: 1. One may get areas by use of the area under curves. 2. The volume computation is mostly about figuring the solid of rotation volume. 3. Applications of physics abound in motion problems, work done, and the center of mass. 4. Calculation of probability density functions makes use of statistics and probability. This extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus. Frequently Asked Questions (FAQ) - What is the use of integration in real life? - Integrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models. - What is the best integral calculator? - Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. - What does to integrate mean? - Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas. Why users love our Integral Calculator \| \| \| \|---\|---\| \| 🌐 Languages \| EN, ES, PT & more \| \| 🏆 Practice \| Improve your math skills \| \| 😍 Step by step \| In depth solution steps \| \| ⭐️ Rating \| [4\.6](https://apps.apple.com/ua/app/symbolab-math-problem-solver/id876942533) based on 20924 reviews \| integral-calculator en Related Symbolab blog posts [Advanced Math Solutions – Integral Calculator, the complete guide We’ve covered quite a few integration techniques, some are straightforward, some are more challenging, but finding...](https://blog.symbolab.com/2015/01/advanced-math-solutions-integral_27.html) Popular topics [scientific calculator](https://www.symbolab.com/solver/scientific-calculator) [inverse calculator](https://www.symbolab.com/solver/function-inverse-calculator) [simplify calculator](https://www.symbolab.com/solver/simplify-calculator) [distance calculator](https://www.symbolab.com/solver/distance-calculator) [fractions calculator](https://www.symbolab.com/solver/fractions-calculator) [interval notation calculator](https://www.symbolab.com/solver/interval-notation-calculator) [cross product calculator](https://www.symbolab.com/solver/vector-cross-product-calculator) [probability calculator](https://www.symbolab.com/solver/probability-calculator) [derivative calculator](https://www.symbolab.com/solver/derivative-calculator) [series calculator](https://www.symbolab.com/solver/series-calculator) [ratios calculator](https://www.symbolab.com/solver/ratios-calculator) [statistics calculator](https://www.symbolab.com/solver/statistics-calculator) [integral calculator](https://www.symbolab.com/solver/integral-calculator) [inverse laplace transform calculator](https://www.symbolab.com/solver/inverse-laplace-calculator) [rounding calculator](https://www.symbolab.com/solver/rounding-calculator) [gcf calculator](https://www.symbolab.com/solver/gcf-calculator) [algebra calculator](https://www.symbolab.com/solver/algebra-calculator) [tangent line calculator](https://www.symbolab.com/solver/tangent-line-calculator) [trigonometry calculator](https://www.symbolab.com/solver/trigonometry-calculator) [log calculator](https://www.symbolab.com/solver/logarithmic-equation-calculator) [standard deviation calculator](https://www.symbolab.com/solver/standard-deviation-calculator) [linear equation calculator](https://www.symbolab.com/solver/linear-equation-calculator) [antiderivative calculator](https://www.symbolab.com/solver/antiderivative-calculator) [laplace transform calculator](https://www.symbolab.com/solver/laplace-calculator) [quadratic equation calculator](https://www.symbolab.com/solver/quadratic-equation-calculator) [domain calculator](https://www.symbolab.com/solver/function-domain-calculator) [decimals calculator](https://www.symbolab.com/solver/decimals-calculator) [limit calculator](https://www.symbolab.com/solver/limit-calculator) [equation solver](https://www.symbolab.com/solver/equation-calculator) [definite integral calculator](https://www.symbolab.com/solver/definite-integral-calculator) [matrix inverse calculator](https://www.symbolab.com/solver/matrix-inverse-calculator) [matrix calculator](https://www.symbolab.com/solver/matrix-calculator) [system of equations calculator](https://www.symbolab.com/solver/system-of-equations-calculator) [calculus calculator](https://www.symbolab.com/solver/calculus-calculator) [slope calculator](https://www.symbolab.com/solver/slope-calculator) [long division calculator](https://www.symbolab.com/solver/polynomial-long-division-calculator) [factors calculator](https://www.symbolab.com/solver/factor-calculator) [polynomial calculator](https://www.symbolab.com/solver/polynomial-equation-calculator) [square root calculator](https://www.symbolab.com/solver/square-root-calculator) [implicit differentiation calculator](https://www.symbolab.com/solver/implicit-derivative-calculator) [word problem solver](https://www.symbolab.com/solver/word-problems-calculator) [differential equation calculator](https://www.symbolab.com/solver/ordinary-differential-equation-calculator) [average calculator](https://www.symbolab.com/calculator/statistics/average) [synthetic division calculator](https://www.symbolab.com/solver/synthetic-division-calculator) Chat with Symbo AI AI may present inaccurate or offensive content that does not represent Symbolab's views. Do not enter any personal information Enter a problem [Cooking Calculators](https://www.symbolab.com/calculator/cooking) [Cooking Measurement Converter](https://www.symbolab.com/calculator/cooking/cooking-measurement) [Cooking Ingredient Converter](https://www.symbolab.com/calculator/cooking/cooking-ingredient) [Cake Pan Converter](https://www.symbolab.com/calculator/cooking/cake-pan) [More calculators](https://www.symbolab.com/calculator/cooking) [Fitness Calculators](https://www.symbolab.com/calculator/fitness) [BMI Calculator](https://www.symbolab.com/calculator/fitness/bmi) [Calorie Calculator](https://www.symbolab.com/calculator/fitness/calorie) [BMR Calculator](https://www.symbolab.com/calculator/fitness/bmr) [More calculators](https://www.symbolab.com/calculator/fitness) Save to Notebook\! Sign in Notebook [View Full Notebook](https://www.symbolab.com/notebook) Study Tools [AI Math Solver](https://www.symbolab.com/) [AI Chat](https://www.symbolab.com/ai-chat) [Popular Problems](https://www.symbolab.com/popular-algebra) [Worksheets](https://www.symbolab.com/worksheets/Pre-Algebra) [Study Guides](https://www.symbolab.com/study-guides) [Practice](https://www.symbolab.com/practice) [Cheat Sheets](https://www.symbolab.com/cheat-sheets) [Calculators](https://www.symbolab.com/calculators) [Graphing Calculator](https://www.symbolab.com/graphing-calculator) [Geometry Calculator](https://www.symbolab.com/geometry-calculator) [Verify Solution](https://www.symbolab.com/verify) Apps [Symbolab App (Android)](https://play.google.com/store/apps/details?id=com.devsense.symbolab) [Graphing Calculator (Android)](https://play.google.com/store/apps/details?id=com.symbolab.graphingcalculator) [Practice (Android)](https://play.google.com/store/apps/details?id=com.symbolab.practice) [Symbolab App (iOS)](https://apps.apple.com/us/app/id876942533) [Graphing Calculator (iOS)](https://apps.apple.com/us/app/id1435182798) [Practice (iOS)](https://apps.apple.com/us/app/id1469186281) [Chrome Extension](https://chromewebstore.google.com/detail/mgfcnnpakbfcefgphceolkjldjfpieff) Company [About Symbolab](https://www.symbolab.com/about) [Blog](https://blog.symbolab.com/) [Help](https://www.symbolab.com/help) [Contact Us](https://www.symbolab.com/solver/integral-calculator) Legal [Privacy](https://www.symbolab.com/privacy-policy) [Service Terms](https://www.symbolab.com/terms-of-use) [Cookie Policy](https://www.symbolab.com/cookie-policy) Cookie Settings Copyright, Community Guidelines, DSA & other Legal Resources [Learneo Legal Center](https://learneo.com/legal) [Feedback](https://www.symbolab.com/solver/integral-calculator) Social Media Symbolab, a Learneo, Inc. business © Learneo, Inc. 2024 * (optional) (optional) Please add a message. Message received. Thanks for the feedback. Cancel Send** ![](https://www.symbolab.com/public/img/ajax-loader.gif) ![]()
Readable Markdown	## Introduction to Integration ### What is an Integration? Integration is the union of elements to create a whole. Integral calculus allows us to find a function whose differential is provided, so integrating is the inverse of differentiating. It defines and computes the area of a region constrained by the [graph of a function](https://www.symbolab.com/graphing-calculator). Integration developed historically from the process of exhaustion, in which inscribing polygons approximated the area of a curved form. We distinguish integration into two forms: [definite](https://www.symbolab.com/solver/definite-integral-calculator) and [indefinite integrals](https://www.symbolab.com/solver/indefinite-integral-calculator). Fundamental instruments in [calculus](https://www.symbolab.com/solver/calculus-calculator), differentiation and integration have extensive use in mathematics and physics. Leibniz created the ideas of integration. Let us investigate integration, its features, and some of its effective approaches. ### Integration - An Inverse Process of Differentiation Integration is the opposite of differentiation basically. Integration helps us to determine the original function of a derivative if provided one. If \$\\frac{d(F(x))}{dx}=f(x)\$, then \$\\int f(x)dx=F(x)+C\$ . This is known as indefinite integrals. For Example Suppose \$f(x)=x^3\$ The [derivative](https://www.symbolab.com/solver/derivative-calculator) of f(x) is \$f'(x)=3x^2\$ The [antiderivative](https://www.symbolab.com/solver/antiderivative-calculator) of \$3x^2\$ is \$x^3\$ So the derivative of any constant is zero and anti-derivative of any expression will contain arbitrary constant denoted by C that is \$ \\int 3x^2dx=x^3+C\$ Therefore, if \$\\frac{dy}{dx}=f(x)\$, then we can write it as y = \$\\int f(x)dx=F(x)+C\$ where: - \$\\int f(x)dx\$ will represent the complete class of integral. - C is an arbitrary constant. - x is the variable of equation. - The symbol \$\\int\$ denotes the integral. - f(x) is the integrand. ### Rules of Integration Sum and Difference Rules: \$\\int \[f(x) + g(x)\] dx = \\int f(x) dx + \\int g(x) dx\$ \$\\int \[f(x) - g(x)\] dx = \\int f(x) dx - \\int g(x) dx\$ For example: \$\\int (x^2 + 3x) dx = \\int x^2 dx + \\int 3x dx\$ \$= \\frac{x^3}{3} + \\frac{3x^2}{2} + C\$ Power Rule: \$ \\int x^ndx= \\frac {x^{(n+1)}}{n+1} +C \$ Please note here n\$\\neq\$-1 For example: \$\\int x^5dx=\\frac{x^6}{6}+C\$ Exponential Rules: \$ \\int e^xdx=e^x+C\$ \$ \\int a^xdx= \\frac{a^x}{ln(a)}+C\$ \$ \\int ln(x)dx=xln(x)-x+C\$ Constant Multiplication Rule: \$ \\int adx=ax+C \$ Reciprocal Rule: \$ \\int \\frac{1}{x}dx=ln\\|x\\|+C \$ ### Properties of Integration Properties of indefinite \$\\int \[f(x) \\pm g(x)\]dx= \\int f(x)dx \\pm \\int g(x)dx \$ \$ \\int kf(x)dx = k \\int f(x)dx \$ (here k is the constant) \$ \\int f(x)dx= \\int g(x)dx \$ if \$ \\int \[f(x)-g(x)\]dx=0 \$ By collabrating these properties, we derive: \$\\int \\sum k \_nf\_n(x)dx=\\sum k\_n\\int f\_n(x)dx \$ ### What are some uses for an integral calculator and what is it exactly? One may find the integrals of functions by use of an integral calculator—a mathematical tool. Solving complex integration problems in a quick and exact way is the main application for this instrument in the domains of education, engineering, and physics. It can manage definite as well as indefinite integrals. ### Implementations of an integral calculator: 1. Two examples of solving definite and indefinite integrals include computing the area under a curve or finding the antiderivative. 2. Double-checking hand computations helps one verify integration answers. 3. "Handling complex functions" is the capacity to combine activities that are challenging for manual handling. 4. Applications in Physics and Engineering: Applied to derive motion equations, work done, and areas under curves by use of these programs. 5. Improving learning means giving students help understanding integration techniques and their uses. For example using an integration calculator we can find: \$\\int (3x^2-2x+1)dx =3\\frac{x^{2+1}}{2+1}-2 \\frac{x^{1+1}}{1+1}+x+C\$ \$=3\\frac{x^{3}}{3}-2 \\frac{x^{2}}{2}+x+C\$ \$=x^3-x^2+x+C\$ ### Methods of Integration The simplest basic search might not always be enough to find an essential. We use several techniques for integration to help to simplify functions into normal forms. The main strategies are listed here: 1\. Integration by Decomposition In this method we need to breakthe function into basic parts: \$\$\\int \\frac{x^2-x+1}{x^3}dx\$\$ Expanding: \$\$\\int (\\frac{x^2}{x^3}-\\frac{x}{x^3}+\\frac{1}{x^3})dx \$\$ Applying the basic rules: \$ log \\|x\\|+ \\frac {1}{x}-\\frac{1}{2x^2}+C\$ 2\. Integration by Substitution For simplyfing the integral change the variables: \$\$ \\int sin(mx)dx \$\$ Let mx=t, so \$dx=dt/m\$ Therefore, \$ \\frac{1}{m} \\int sin t dt= -\\frac{1}{m}cos t +C \$ 3\. Integration using Partial Fractions We use this function for rational functions: \$ \\int \\frac{1}{(x+1)(x+2)}dx=\\frac{A}{x+1}+ \\frac{B}{x+2} \$ so now we can integrate them seperately and solve for the value of A and B. 4\. Integration by Parts Derived from the product rule of differentiation: \$\$ \\int udv = uv - \\int vdu \$\$ ### Applications of Integration Integration has great use in domains like physics, engineering, and economics. Of the greatest significance are: 1. One may get areas by use of the area under curves. 2. The volume computation is mostly about figuring the solid of rotation volume. 3. Applications of physics abound in motion problems, work done, and the center of mass. 4. Calculation of probability density functions makes use of statistics and probability. This extensive reference to integration addresses its basic ideas, guidelines, and methods, which provide the foundation for more complex uses of calculus. Frequently Asked Questions (FAQ) - What is the use of integration in real life? - Integrations is used in various fields such as engineering to determine the shape and size of strcutures. In Physics to find the centre of gravity. In the field of graphical representation to build three-dimensional models. - What is the best integral calculator? - Symbolab is the best integral calculator solving indefinite integrals, definite integrals, improper integrals, double integrals, triple integrals, multiple integrals, antiderivatives, and more. - What does to integrate mean? - Integration is a way to sum up parts to find the whole. It is used to find the area under a curve by slicing it to small rectangles and summing up thier areas.
Shard	2 (laksa)
Root Hash	1977249391401099202
Unparsed URL	com,symbolab!www,/solver/integral-calculator s443