🕷️ Crawler Inspector

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Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 166 (from laksa117)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa166.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.wolframalpha.com/calculators/integral-calculator/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.wolframalpha.com/calculators/integral-calculator/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.wolframalpha.com\/calculators\/integral-calculator\/","crawl_time":1776037968,"first_indexed_time":1401649530,"http_code":200,"src_unparsed":"com,wolframalpha!www,\/calculators\/integral-calculator\/ s443","src_root_hash":"10472823281220336366","history_drop_reason":null,"meta_title":"Integral Calculator: Step-by-Step Solutions - Wolfram|Alpha","meta_descriptions":["Free Integral Calculator helps you solve definite and indefinite integration problems. Also double, triple and improper integrals. Answers, graphs, alternate forms."],"attrs_boilerpipe_text":"WolframAlpha\nSolve integrals with Wolfram|Alpha\nx\ns\ni\nn\nx\n2\nd\nx\nMath Input\nMore than just an online integral solver\nWolfram|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition.\nLearn more about:\nIntegrals\n »\nTips for entering queries\nUse Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English.\nintegrate x\/(x-1)\nintegrate x sin(x^2)\nintegrate x sqrt(1-sqrt(x))\nintegrate x\/(x+1)^3 from 0 to infinity\nintegrate 1\/(cos(x)+2) from 0 to 2pi\nintegrate x^2 sin y dx dy, x=0 to 1, y=0 to pi\nView more examples\n »\nAccess instant learning tools\nGet immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator\nLearn more about:\nStep-by-step solutions\n »\nWolfram Problem Generator\n »\nVIEW ALL CALCULATORS\nBMI Calculator\nDilution Calculator\nMortgage Calculator\nInterest Calculator\nLoan Calculator\nPresent Value Calculator\nCar Payment Calculator\nFuture Value Calculator\nLimit Calculator\nDerivative Calculator\nDouble Integral Calculator\nTriple Integral Calculator\nSeries Expansion Calculator\nDiscontinuity Calculator\nDomain and Range Calculator\nFactoring Calculator\nQuadratic Formula Calculator\nEquation Solver Calculator\nPartial Fraction Decomposition Calculator\nSystem of Equations Calculator\nDeterminant Calculator\nEigenvalue Calculator\nMatrix Inverse Calculator\nWhat are integrals?\nIntegration is an important tool in calculus that can give an antiderivative or represent area under a curve.\nThe indefinite integral of\nf (x)\nf\nx\n, denoted\nStart  Integral, Start integrand,  f (x) , integrand End,Start first variable,  x , first variable End ,  Integral End\nf\nx\nd\nx\n, is defined to be the antiderivative of\nf (x)\nf\nx\n. In other words, the derivative of\nStart  Integral, Start integrand,  f (x) , integrand End,Start first variable,  x , first variable End ,  Integral End\nf\nx\nd\nx\nis\nf (x)\nf\nx\n. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,\nStart  Integral, Start integrand,  Start Sine, Start angle,  x , angle End , Sine End , integrand End,Start first variable,  x , first variable End ,  Integral End  = - Start Cosine, Start angle,  x , angle End , Cosine End +constant\nsin\nx\nd\nx\n=\n-\ncos\nx\n+\nc\no\nn\ns\nt\na\nn\nt\n, since the derivative of\n− Start Cosine, Start angle,  x , angle End , Cosine End  + constant\n−\ncos\nx\n+\nc\no\nn\ns\nt\na\nn\nt\nis\nStart Sine, Start angle,  x , angle End , Sine End\nsin\nx\n. The definite integral of\nf (x)\nf\nx\nfrom\nx = a\nx\n=\na\nto\nx = b\nx\n=\nb\n, denoted\nStart Definite  Integral, Start first lower limit,  a , first lower limit End,Start first upper limit,  b , first upper limit End,Start integrand,  f (x) , integrand End,Start first variable,  x , first variable End , Definite  Integral End\nb\na\nf\nx\nd\nx\n, is defined to be the signed area between\nf (x)\nf\nx\nand the\nx\nx\naxis, from\nx = a\nx\n=\na\nto\nx = b\nx\n=\nb\n.\nBoth types of integrals are tied together by the fundamental theorem of calculus. This states that if\nf (x)\nf\nx\nis continuous on\n[a,b]\na\n,\nb\nand\nF(x)\nF\nx\nis its continuous indefinite integral, then\nStart Definite  Integral, Start first lower limit,  a , first lower limit End,Start first upper limit,  b , first upper limit End,Start integrand,  f (x) , integrand End,Start first variable,  x , first variable End , Definite  Integral End  = F(b) - F(a)\nb\na\nf\nx\nd\nx\n=\nF\nb\n-\nF\na\n. This means\nStart Definite  Integral, Start first lower limit,  0 , first lower limit End,Start first upper limit,  π , first upper limit End,Start integrand,  Start Sine, Start angle,  x , angle End , Sine End , integrand End,Start first variable,  x , first variable End , Definite  Integral End  = (- Start Cosine, Start angle,  π , angle End , Cosine End ) - (- Start Cosine, Start angle,  0 , angle End , Cosine End ) = 2\nπ\n0\nsin\nx\nd\nx\n=\n-\ncos\nπ\n-\n-\ncos\n0\n=\n2\n. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram|Alpha can\nsolve a broad range of integrals\n.\n.\nHow Wolfram|Alpha calculates integrals\nWolfram|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions.\nWhile these powerful algorithms give Wolfram|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions.","attrs_markdown":"![no-js-running-man-logo](https:\/\/www.wolframalpha.com\/_next\/static\/images\/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript.\n\nPlease enable JavaScript. If you don't know how, you can find instructions[here](https:\/\/enable-javascript.com\/).Once you've done that, refresh this page to start using Wolfram\\|Alpha.\n\nWolframAlpha\n\n# Online Integral Calculator\n## Solve integrals with Wolfram\\|Alpha\n\n- Natural Language\n- Math Input\n\n## More than just an online integral solver\nWolfram\\|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram\\|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition.\n\n![Integral results with plots, alternate forms, series expansions and answers](https:\/\/www.wolframalpha.com\/_next\/static\/images\/Integral-Data-image-small_31b2db7afd4584a4573378ea4a1140ac.png)\n\nLearn more about:\n\n- [Integrals »](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/)\n\n## Tips for entering queries\nUse Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English.\n\n- [integrate x\/(x-1)](https:\/\/www.wolframalpha.com\/input?i=integrate+x%2F%28x-1%29)\n- [integrate x sin(x^2)](https:\/\/www.wolframalpha.com\/input?i=integrate+x+sin%28x%5E2%29)\n- [integrate x sqrt(1-sqrt(x))](https:\/\/www.wolframalpha.com\/input?i=integrate+x+sqrt%281-sqrt%28x%29%29)\n\n- [integrate x\/(x+1)^3 from 0 to infinity](https:\/\/www.wolframalpha.com\/input?i=integrate+x%2F%28x%2B1%29%5E3+from+0+to+infinity)\n- [integrate 1\/(cos(x)+2) from 0 to 2pi](https:\/\/www.wolframalpha.com\/input?i=integrate+1%2F%28cos%28x%29%2B2%29+from+0+to+2pi)\n- [integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi](https:\/\/www.wolframalpha.com\/input?i=integrate+x%5E2+sin+y+dx+dy%2C+x%3D0+to+1%2C+y%3D0+to+pi)\n\n- [View more examples »](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/)\n\n## Access instant learning tools\nGet immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator\n\n![Step-by-step solutions for integrals with detailed breakdowns and unlimited Wolfram Problem Generator eigenvalue practice problems](https:\/\/www.wolframalpha.com\/_next\/static\/images\/Integral-Learning-Tools-small_840fc3d452038a7d147f04090c7b45a7.png)\n\nLearn more about:\n\n- [Step-by-step solutions »](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus\/)\n- [Wolfram Problem Generator »](https:\/\/www.wolframalpha.com\/problem-generator\/?nonexternal=true&scrollTo=Calculus)\n\n## ![Spikey](https:\/\/www.wolframalpha.com\/_next\/static\/images\/alpha_small_spikey_342886801177880d3bb9790a29a3b07c.png)VIEW ALL CALCULATORS\n- [BMI Calculator](https:\/\/www.wolframalpha.com\/calculators\/bmi-calculator)\n- [Dilution Calculator](https:\/\/www.wolframalpha.com\/calculators\/dilution-calculator)\n- [Mortgage Calculator](https:\/\/www.wolframalpha.com\/calculators\/mortgage-calculator)\n- [Interest Calculator](https:\/\/www.wolframalpha.com\/calculators\/interest-calculator)\n- [Loan Calculator](https:\/\/www.wolframalpha.com\/calculators\/loan-calculator)\n- [Present Value Calculator](https:\/\/www.wolframalpha.com\/calculators\/present-value-calculator)\n- [Car Payment Calculator](https:\/\/www.wolframalpha.com\/calculators\/car-payment-calculator)\n- [Future Value Calculator](https:\/\/www.wolframalpha.com\/calculators\/future-value-calculator)\n- [Limit Calculator](https:\/\/www.wolframalpha.com\/calculators\/limit-calculator)\n- [Derivative Calculator](https:\/\/www.wolframalpha.com\/calculators\/derivative-calculator)\n- [Double Integral Calculator](https:\/\/www.wolframalpha.com\/calculators\/double-integral-calculator)\n- [Triple Integral Calculator](https:\/\/www.wolframalpha.com\/calculators\/triple-integral-calculator)\n- [Series Expansion Calculator](https:\/\/www.wolframalpha.com\/calculators\/series-calculator)\n- [Discontinuity Calculator](https:\/\/www.wolframalpha.com\/calculators\/discontinuity-calculator)\n- [Domain and Range Calculator](https:\/\/www.wolframalpha.com\/calculators\/domain-range-calculator)\n- [Factoring Calculator](https:\/\/www.wolframalpha.com\/calculators\/factoring-calculator)\n- [Quadratic Formula Calculator](https:\/\/www.wolframalpha.com\/calculators\/quadratic-calculator)\n- [Equation Solver Calculator](https:\/\/www.wolframalpha.com\/calculators\/equation-solver-calculator)\n- [Partial Fraction Decomposition Calculator](https:\/\/www.wolframalpha.com\/calculators\/partial-fraction-calculator)\n- [System of Equations Calculator](https:\/\/www.wolframalpha.com\/calculators\/system-equation-calculator)\n- [Determinant Calculator](https:\/\/www.wolframalpha.com\/calculators\/determinant-calculator)\n- [Eigenvalue Calculator](https:\/\/www.wolframalpha.com\/calculators\/eigenvalue-calculator)\n- [Matrix Inverse Calculator](https:\/\/www.wolframalpha.com\/calculators\/matrix-inverse-calculator)\n\n## What are integrals?\n### Integration is an important tool in calculus that can give an antiderivative or represent area under a curve.\nThe indefinite integral of f (x) f x, denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx, is defined to be the antiderivative of f (x) f x. In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx is f (x) f x. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constant sin x dx \\= \\- cos x \\+constant, since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant − cos x \\+ constant is Start Sine, Start angle, x , angle End , Sine End sin x. The definite integral of f (x) f x from x = a x \\= a to x = b x \\= b, denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End ba f x dx, is defined to be the signed area between f (x) f x and the x x axis, from x = a x \\= a to x = b x \\= b.\n\nBoth types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x) f x is continuous on \\[a,b\\] a,b and F(x) F x is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a) ba f x dx \\= F b \\- F a. This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2 π0 sin x dx \\= \\- cos π \\- \\- cos 0 \\= 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram\\|Alpha can [solve a broad range of integrals](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/). .\n\n## How Wolfram\\|Alpha calculates integrals\nWolfram\\|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions.\n\nWhile these powerful algorithms give Wolfram\\|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram\\|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions.\n\n- [Pro](https:\/\/www.wolframalpha.com\/pro)\n- [Mobile Apps](https:\/\/products.wolframalpha.com\/mobile)\n- [Products](https:\/\/products.wolframalpha.com\/)\n- [Business](https:\/\/products.wolframalpha.com\/enterprise)\n- [API & Developer Solutions](https:\/\/products.wolframalpha.com\/api)\n- [LLM Solutions](https:\/\/www.wolfram.com\/resources\/tools-for-AIs\/)\n\n- [Resources & Tools](https:\/\/www.wolframalpha.com\/resources)\n- [About](https:\/\/www.wolframalpha.com\/about)\n- [Contact](https:\/\/www.wolframalpha.com\/contact-us)\n- [Connect](https:\/\/www.wolframalpha.com\/social)\n\n- ©2026 Wolfram\n- [Terms](https:\/\/www.wolframalpha.com\/termsofuse)\n- [Privacy](https:\/\/www.wolfram.com\/legal\/privacy\/wolfram)\n\n- [wolfram.com](https:\/\/www.wolfram.com\/)\n- [Wolfram Language](https:\/\/www.wolfram.com\/language)\n- [Mathematica](https:\/\/www.wolfram.com\/mathematica)\n- [Wolfram Demonstrations](https:\/\/demonstrations.wolfram.com\/)\n- [Wolfram for Education](https:\/\/www.wolfram.com\/education)\n- [MathWorld](https:\/\/mathworld.wolfram.com\/)","attrs_readable_markdown":"WolframAlpha\n\n## Solve integrals with Wolfram\\|Alpha\n\n## More than just an online integral solver\nWolfram\\|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram\\|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition.\n\n![Integral results with plots, alternate forms, series expansions and answers](https:\/\/www.wolframalpha.com\/_next\/static\/images\/Integral-Data-image-small_31b2db7afd4584a4573378ea4a1140ac.png)\n\nLearn more about:\n\n- [Integrals »](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/)\n\n## Tips for entering queries\nUse Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English.\n\n- [integrate x\/(x-1)](https:\/\/www.wolframalpha.com\/input?i=integrate+x%2F%28x-1%29)\n- [integrate x sin(x^2)](https:\/\/www.wolframalpha.com\/input?i=integrate+x+sin%28x%5E2%29)\n- [integrate x sqrt(1-sqrt(x))](https:\/\/www.wolframalpha.com\/input?i=integrate+x+sqrt%281-sqrt%28x%29%29)\n\n- [integrate x\/(x+1)^3 from 0 to infinity](https:\/\/www.wolframalpha.com\/input?i=integrate+x%2F%28x%2B1%29%5E3+from+0+to+infinity)\n- [integrate 1\/(cos(x)+2) from 0 to 2pi](https:\/\/www.wolframalpha.com\/input?i=integrate+1%2F%28cos%28x%29%2B2%29+from+0+to+2pi)\n- [integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi](https:\/\/www.wolframalpha.com\/input?i=integrate+x%5E2+sin+y+dx+dy%2C+x%3D0+to+1%2C+y%3D0+to+pi)\n\n- [View more examples »](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/)\n\n## Access instant learning tools\nGet immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator\n\n![Step-by-step solutions for integrals with detailed breakdowns and unlimited Wolfram Problem Generator eigenvalue practice problems](https:\/\/www.wolframalpha.com\/_next\/static\/images\/Integral-Learning-Tools-small_840fc3d452038a7d147f04090c7b45a7.png)\n\nLearn more about:\n\n- [Step-by-step solutions »](https:\/\/www.wolframalpha.com\/examples\/pro-features\/step-by-step-solutions\/step-by-step-calculus\/)\n- [Wolfram Problem Generator »](https:\/\/www.wolframalpha.com\/problem-generator\/?nonexternal=true&scrollTo=Calculus)\n\n## ![Spikey](https:\/\/www.wolframalpha.com\/_next\/static\/images\/alpha_small_spikey_342886801177880d3bb9790a29a3b07c.png)VIEW ALL CALCULATORS\n- [BMI Calculator](https:\/\/www.wolframalpha.com\/calculators\/bmi-calculator)\n- [Dilution Calculator](https:\/\/www.wolframalpha.com\/calculators\/dilution-calculator)\n- [Mortgage Calculator](https:\/\/www.wolframalpha.com\/calculators\/mortgage-calculator)\n- [Interest Calculator](https:\/\/www.wolframalpha.com\/calculators\/interest-calculator)\n- [Loan Calculator](https:\/\/www.wolframalpha.com\/calculators\/loan-calculator)\n- [Present Value Calculator](https:\/\/www.wolframalpha.com\/calculators\/present-value-calculator)\n- [Car Payment Calculator](https:\/\/www.wolframalpha.com\/calculators\/car-payment-calculator)\n- [Future Value Calculator](https:\/\/www.wolframalpha.com\/calculators\/future-value-calculator)\n- [Limit Calculator](https:\/\/www.wolframalpha.com\/calculators\/limit-calculator)\n- [Derivative Calculator](https:\/\/www.wolframalpha.com\/calculators\/derivative-calculator)\n- [Double Integral Calculator](https:\/\/www.wolframalpha.com\/calculators\/double-integral-calculator)\n- [Triple Integral Calculator](https:\/\/www.wolframalpha.com\/calculators\/triple-integral-calculator)\n- [Series Expansion Calculator](https:\/\/www.wolframalpha.com\/calculators\/series-calculator)\n- [Discontinuity Calculator](https:\/\/www.wolframalpha.com\/calculators\/discontinuity-calculator)\n- [Domain and Range Calculator](https:\/\/www.wolframalpha.com\/calculators\/domain-range-calculator)\n- [Factoring Calculator](https:\/\/www.wolframalpha.com\/calculators\/factoring-calculator)\n- [Quadratic Formula Calculator](https:\/\/www.wolframalpha.com\/calculators\/quadratic-calculator)\n- [Equation Solver Calculator](https:\/\/www.wolframalpha.com\/calculators\/equation-solver-calculator)\n- [Partial Fraction Decomposition Calculator](https:\/\/www.wolframalpha.com\/calculators\/partial-fraction-calculator)\n- [System of Equations Calculator](https:\/\/www.wolframalpha.com\/calculators\/system-equation-calculator)\n- [Determinant Calculator](https:\/\/www.wolframalpha.com\/calculators\/determinant-calculator)\n- [Eigenvalue Calculator](https:\/\/www.wolframalpha.com\/calculators\/eigenvalue-calculator)\n- [Matrix Inverse Calculator](https:\/\/www.wolframalpha.com\/calculators\/matrix-inverse-calculator)\n\n## What are integrals?\n### Integration is an important tool in calculus that can give an antiderivative or represent area under a curve.\nThe indefinite integral of f (x) f x, denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx, is defined to be the antiderivative of f (x) f x. In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx is f (x) f x. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constant sin x dx \\= \\- cos x \\+constant, since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant − cos x \\+ constant is Start Sine, Start angle, x , angle End , Sine End sin x. The definite integral of f (x) f x from x = a x \\= a to x = b x \\= b, denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End ba f x dx, is defined to be the signed area between f (x) f x and the x x axis, from x = a x \\= a to x = b x \\= b.\n\nBoth types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x) f x is continuous on \\[a,b\\] a,b and F(x) F x is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a) ba f x dx \\= F b \\- F a. This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2 π0 sin x dx \\= \\- cos π \\- \\- cos 0 \\= 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram\\|Alpha can [solve a broad range of integrals](https:\/\/www.wolframalpha.com\/examples\/mathematics\/calculus-and-analysis\/integrals\/). .\n\n## How Wolfram\\|Alpha calculates integrals\nWolfram\\|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions.\n\nWhile these powerful algorithms give Wolfram\\|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram\\|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions.","meta_canonical":null}

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URL	https://www.wolframalpha.com/calculators/integral-calculator/
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Meta Title	Integral Calculator: Step-by-Step Solutions - Wolfram\|Alpha
Meta Description	Free Integral Calculator helps you solve definite and indefinite integration problems. Also double, triple and improper integrals. Answers, graphs, alternate forms.
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Boilerpipe Text	WolframAlpha Solve integrals with Wolfram\|Alpha x s i n x 2 d x Math Input More than just an online integral solver Wolfram\|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram\|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition. Learn more about: Integrals » Tips for entering queries Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English. integrate x/(x-1) integrate x sin(x^2) integrate x sqrt(1-sqrt(x)) integrate x/(x+1)^3 from 0 to infinity integrate 1/(cos(x)+2) from 0 to 2pi integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi View more examples » Access instant learning tools Get immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator Learn more about: Step-by-step solutions » Wolfram Problem Generator » VIEW ALL CALCULATORS BMI Calculator Dilution Calculator Mortgage Calculator Interest Calculator Loan Calculator Present Value Calculator Car Payment Calculator Future Value Calculator Limit Calculator Derivative Calculator Double Integral Calculator Triple Integral Calculator Series Expansion Calculator Discontinuity Calculator Domain and Range Calculator Factoring Calculator Quadratic Formula Calculator Equation Solver Calculator Partial Fraction Decomposition Calculator System of Equations Calculator Determinant Calculator Eigenvalue Calculator Matrix Inverse Calculator What are integrals? Integration is an important tool in calculus that can give an antiderivative or represent area under a curve. The indefinite integral of f (x) f x , denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x d x , is defined to be the antiderivative of f (x) f x . In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x d x is f (x) f x . Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example, Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constant sin x d x = - cos x + c o n s t a n t , since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant − cos x + c o n s t a n t is Start Sine, Start angle, x , angle End , Sine End sin x . The definite integral of f (x) f x from x = a x = a to x = b x = b , denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End b a f x d x , is defined to be the signed area between f (x) f x and the x x axis, from x = a x = a to x = b x = b . Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x) f x is continuous on [a,b] a , b and F(x) F x is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a) b a f x d x = F b - F a . This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2 π 0 sin x d x = - cos π - - cos 0 = 2 . Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram\|Alpha can solve a broad range of integrals . . How Wolfram\|Alpha calculates integrals Wolfram\|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions. While these powerful algorithms give Wolfram\|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram\|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions.
Markdown	![no-js-running-man-logo](https://www.wolframalpha.com/_next/static/images/running-man_d23cee5b24bc562c49a0355e9408ad8c.png)Uh oh! Wolfram\\|Alpha doesn't run without JavaScript. Please enable JavaScript. If you don't know how, you can find instructions[here](https://enable-javascript.com/).Once you've done that, refresh this page to start using Wolfram\\|Alpha. WolframAlpha # Online Integral Calculator ## Solve integrals with Wolfram\\|Alpha - Natural Language - Math Input ## More than just an online integral solver Wolfram\\|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram\\|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition. ![Integral results with plots, alternate forms, series expansions and answers](https://www.wolframalpha.com/_next/static/images/Integral-Data-image-small_31b2db7afd4584a4573378ea4a1140ac.png) Learn more about: - [Integrals »](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/) ## Tips for entering queries Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English. - [integrate x/(x-1)](https://www.wolframalpha.com/input?i=integrate+x%2F%28x-1%29) - [integrate x sin(x^2)](https://www.wolframalpha.com/input?i=integrate+x+sin%28x%5E2%29) - [integrate x sqrt(1-sqrt(x))](https://www.wolframalpha.com/input?i=integrate+x+sqrt%281-sqrt%28x%29%29) - [integrate x/(x+1)^3 from 0 to infinity](https://www.wolframalpha.com/input?i=integrate+x%2F%28x%2B1%29%5E3+from+0+to+infinity) - [integrate 1/(cos(x)+2) from 0 to 2pi](https://www.wolframalpha.com/input?i=integrate+1%2F%28cos%28x%29%2B2%29+from+0+to+2pi) - [integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi](https://www.wolframalpha.com/input?i=integrate+x%5E2+sin+y+dx+dy%2C+x%3D0+to+1%2C+y%3D0+to+pi) - [View more examples »](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/) ## Access instant learning tools Get immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator ![Step-by-step solutions for integrals with detailed breakdowns and unlimited Wolfram Problem Generator eigenvalue practice problems](https://www.wolframalpha.com/_next/static/images/Integral-Learning-Tools-small_840fc3d452038a7d147f04090c7b45a7.png) Learn more about: - [Step-by-step solutions »](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus/) - [Wolfram Problem Generator »](https://www.wolframalpha.com/problem-generator/?nonexternal=true&scrollTo=Calculus) ## ![Spikey](https://www.wolframalpha.com/_next/static/images/alpha_small_spikey_342886801177880d3bb9790a29a3b07c.png)VIEW ALL CALCULATORS - [BMI Calculator](https://www.wolframalpha.com/calculators/bmi-calculator) - [Dilution Calculator](https://www.wolframalpha.com/calculators/dilution-calculator) - [Mortgage Calculator](https://www.wolframalpha.com/calculators/mortgage-calculator) - [Interest Calculator](https://www.wolframalpha.com/calculators/interest-calculator) - [Loan Calculator](https://www.wolframalpha.com/calculators/loan-calculator) - [Present Value Calculator](https://www.wolframalpha.com/calculators/present-value-calculator) - [Car Payment Calculator](https://www.wolframalpha.com/calculators/car-payment-calculator) - [Future Value Calculator](https://www.wolframalpha.com/calculators/future-value-calculator) - [Limit Calculator](https://www.wolframalpha.com/calculators/limit-calculator) - [Derivative Calculator](https://www.wolframalpha.com/calculators/derivative-calculator) - [Double Integral Calculator](https://www.wolframalpha.com/calculators/double-integral-calculator) - [Triple Integral Calculator](https://www.wolframalpha.com/calculators/triple-integral-calculator) - [Series Expansion Calculator](https://www.wolframalpha.com/calculators/series-calculator) - [Discontinuity Calculator](https://www.wolframalpha.com/calculators/discontinuity-calculator) - [Domain and Range Calculator](https://www.wolframalpha.com/calculators/domain-range-calculator) - [Factoring Calculator](https://www.wolframalpha.com/calculators/factoring-calculator) - [Quadratic Formula Calculator](https://www.wolframalpha.com/calculators/quadratic-calculator) - [Equation Solver Calculator](https://www.wolframalpha.com/calculators/equation-solver-calculator) - [Partial Fraction Decomposition Calculator](https://www.wolframalpha.com/calculators/partial-fraction-calculator) - [System of Equations Calculator](https://www.wolframalpha.com/calculators/system-equation-calculator) - [Determinant Calculator](https://www.wolframalpha.com/calculators/determinant-calculator) - [Eigenvalue Calculator](https://www.wolframalpha.com/calculators/eigenvalue-calculator) - [Matrix Inverse Calculator](https://www.wolframalpha.com/calculators/matrix-inverse-calculator) ## What are integrals? ### Integration is an important tool in calculus that can give an antiderivative or represent area under a curve. The indefinite integral of f (x) f x, denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx, is defined to be the antiderivative of f (x) f x. In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx is f (x) f x. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constant sin x dx \= \- cos x \+constant, since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant − cos x \+ constant is Start Sine, Start angle, x , angle End , Sine End sin x. The definite integral of f (x) f x from x = a x \= a to x = b x \= b, denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End ba f x dx, is defined to be the signed area between f (x) f x and the x x axis, from x = a x \= a to x = b x \= b. Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x) f x is continuous on \[a,b\] a,b and F(x) F x is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a) ba f x dx \= F b \- F a. This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2 π0 sin x dx \= \- cos π \- \- cos 0 \= 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram\\|Alpha can [solve a broad range of integrals](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/). . ## How Wolfram\\|Alpha calculates integrals Wolfram\\|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions. While these powerful algorithms give Wolfram\\|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram\\|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions. - [Pro](https://www.wolframalpha.com/pro) - [Mobile Apps](https://products.wolframalpha.com/mobile) - [Products](https://products.wolframalpha.com/) - [Business](https://products.wolframalpha.com/enterprise) - [API & Developer Solutions](https://products.wolframalpha.com/api) - [LLM Solutions](https://www.wolfram.com/resources/tools-for-AIs/) - [Resources & Tools](https://www.wolframalpha.com/resources) - [About](https://www.wolframalpha.com/about) - [Contact](https://www.wolframalpha.com/contact-us) - [Connect](https://www.wolframalpha.com/social) - ©2026 Wolfram - [Terms](https://www.wolframalpha.com/termsofuse) - [Privacy](https://www.wolfram.com/legal/privacy/wolfram) - [wolfram.com](https://www.wolfram.com/) - [Wolfram Language](https://www.wolfram.com/language) - [Mathematica](https://www.wolfram.com/mathematica) - [Wolfram Demonstrations](https://demonstrations.wolfram.com/) - [Wolfram for Education](https://www.wolfram.com/education) - [MathWorld](https://mathworld.wolfram.com/)
Readable Markdown	WolframAlpha ## Solve integrals with Wolfram\\|Alpha ## More than just an online integral solver Wolfram\\|Alpha is a great tool for calculating antiderivatives and definite integrals, double and triple integrals, and improper integrals. The Wolfram\\|Alpha Integral Calculator also shows plots, alternate forms and other relevant information to enhance your mathematical intuition. ![Integral results with plots, alternate forms, series expansions and answers](https://www.wolframalpha.com/_next/static/images/Integral-Data-image-small_31b2db7afd4584a4573378ea4a1140ac.png) Learn more about: - [Integrals »](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/) ## Tips for entering queries Use Math Input above or enter your integral calculator queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask for an integral using plain English. - [integrate x/(x-1)](https://www.wolframalpha.com/input?i=integrate+x%2F%28x-1%29) - [integrate x sin(x^2)](https://www.wolframalpha.com/input?i=integrate+x+sin%28x%5E2%29) - [integrate x sqrt(1-sqrt(x))](https://www.wolframalpha.com/input?i=integrate+x+sqrt%281-sqrt%28x%29%29) - [integrate x/(x+1)^3 from 0 to infinity](https://www.wolframalpha.com/input?i=integrate+x%2F%28x%2B1%29%5E3+from+0+to+infinity) - [integrate 1/(cos(x)+2) from 0 to 2pi](https://www.wolframalpha.com/input?i=integrate+1%2F%28cos%28x%29%2B2%29+from+0+to+2pi) - [integrate x^2 sin y dx dy, x=0 to 1, y=0 to pi](https://www.wolframalpha.com/input?i=integrate+x%5E2+sin+y+dx+dy%2C+x%3D0+to+1%2C+y%3D0+to+pi) - [View more examples »](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/) ## Access instant learning tools Get immediate feedback and guidance with step-by-step solutions for integrals and Wolfram Problem Generator ![Step-by-step solutions for integrals with detailed breakdowns and unlimited Wolfram Problem Generator eigenvalue practice problems](https://www.wolframalpha.com/_next/static/images/Integral-Learning-Tools-small_840fc3d452038a7d147f04090c7b45a7.png) Learn more about: - [Step-by-step solutions »](https://www.wolframalpha.com/examples/pro-features/step-by-step-solutions/step-by-step-calculus/) - [Wolfram Problem Generator »](https://www.wolframalpha.com/problem-generator/?nonexternal=true&scrollTo=Calculus) ## ![Spikey](https://www.wolframalpha.com/_next/static/images/alpha_small_spikey_342886801177880d3bb9790a29a3b07c.png)VIEW ALL CALCULATORS - [BMI Calculator](https://www.wolframalpha.com/calculators/bmi-calculator) - [Dilution Calculator](https://www.wolframalpha.com/calculators/dilution-calculator) - [Mortgage Calculator](https://www.wolframalpha.com/calculators/mortgage-calculator) - [Interest Calculator](https://www.wolframalpha.com/calculators/interest-calculator) - [Loan Calculator](https://www.wolframalpha.com/calculators/loan-calculator) - [Present Value Calculator](https://www.wolframalpha.com/calculators/present-value-calculator) - [Car Payment Calculator](https://www.wolframalpha.com/calculators/car-payment-calculator) - [Future Value Calculator](https://www.wolframalpha.com/calculators/future-value-calculator) - [Limit Calculator](https://www.wolframalpha.com/calculators/limit-calculator) - [Derivative Calculator](https://www.wolframalpha.com/calculators/derivative-calculator) - [Double Integral Calculator](https://www.wolframalpha.com/calculators/double-integral-calculator) - [Triple Integral Calculator](https://www.wolframalpha.com/calculators/triple-integral-calculator) - [Series Expansion Calculator](https://www.wolframalpha.com/calculators/series-calculator) - [Discontinuity Calculator](https://www.wolframalpha.com/calculators/discontinuity-calculator) - [Domain and Range Calculator](https://www.wolframalpha.com/calculators/domain-range-calculator) - [Factoring Calculator](https://www.wolframalpha.com/calculators/factoring-calculator) - [Quadratic Formula Calculator](https://www.wolframalpha.com/calculators/quadratic-calculator) - [Equation Solver Calculator](https://www.wolframalpha.com/calculators/equation-solver-calculator) - [Partial Fraction Decomposition Calculator](https://www.wolframalpha.com/calculators/partial-fraction-calculator) - [System of Equations Calculator](https://www.wolframalpha.com/calculators/system-equation-calculator) - [Determinant Calculator](https://www.wolframalpha.com/calculators/determinant-calculator) - [Eigenvalue Calculator](https://www.wolframalpha.com/calculators/eigenvalue-calculator) - [Matrix Inverse Calculator](https://www.wolframalpha.com/calculators/matrix-inverse-calculator) ## What are integrals? ### Integration is an important tool in calculus that can give an antiderivative or represent area under a curve. The indefinite integral of f (x) f x, denoted Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx, is defined to be the antiderivative of f (x) f x. In other words, the derivative of Start Integral, Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Integral End f x dx is f (x) f x. Since the derivative of a constant is 0, indefinite integrals are defined only up to an arbitrary constant. For example,Start Integral, Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Integral End = - Start Cosine, Start angle, x , angle End , Cosine End +constant sin x dx \= \- cos x \+constant, since the derivative of − Start Cosine, Start angle, x , angle End , Cosine End + constant − cos x \+ constant is Start Sine, Start angle, x , angle End , Sine End sin x. The definite integral of f (x) f x from x = a x \= a to x = b x \= b, denoted Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End ba f x dx, is defined to be the signed area between f (x) f x and the x x axis, from x = a x \= a to x = b x \= b. Both types of integrals are tied together by the fundamental theorem of calculus. This states that if f (x) f x is continuous on \[a,b\] a,b and F(x) F x is its continuous indefinite integral, then Start Definite Integral, Start first lower limit, a , first lower limit End,Start first upper limit, b , first upper limit End,Start integrand, f (x) , integrand End,Start first variable, x , first variable End , Definite Integral End = F(b) - F(a) ba f x dx \= F b \- F a. This means Start Definite Integral, Start first lower limit, 0 , first lower limit End,Start first upper limit, π , first upper limit End,Start integrand, Start Sine, Start angle, x , angle End , Sine End , integrand End,Start first variable, x , first variable End , Definite Integral End = (- Start Cosine, Start angle, π , angle End , Cosine End ) - (- Start Cosine, Start angle, 0 , angle End , Cosine End ) = 2 π0 sin x dx \= \- cos π \- \- cos 0 \= 2. Sometimes an approximation to a definite integral is desired. A common way to do so is to place thin rectangles under the curve and add the signed areas together. Wolfram\\|Alpha can [solve a broad range of integrals](https://www.wolframalpha.com/examples/mathematics/calculus-and-analysis/integrals/). . ## How Wolfram\\|Alpha calculates integrals Wolfram\\|Alpha computes integrals differently than people. It calls Mathematica's Integrate function, which represents a huge amount of mathematical and computational research. Integrate does not do integrals the way people do. Instead, it uses powerful, general algorithms that often involve very sophisticated math. There are a couple of approaches that it most commonly takes. One involves working out the general form for an integral, then differentiating this form and solving equations to match undetermined symbolic parameters. Even for quite simple integrands, the equations generated in this way can be highly complex and require Mathematica's strong algebraic computation capabilities to solve. Another approach that Mathematica uses in working out integrals is to convert them to generalized hypergeometric functions, then use collections of relations about these highly general mathematical functions. While these powerful algorithms give Wolfram\\|Alpha the ability to compute integrals very quickly and handle a wide array of special functions, understanding how a human would integrate is important too. As a result, Wolfram\\|Alpha also has algorithms to perform integrations step by step. These use completely different integration techniques that mimic the way humans would approach an integral. This includes integration by substitution, integration by parts, trigonometric substitution and integration by partial fractions.
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