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Calculated Shard: 189 (from laksa090)

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curl -X POST \
  'http://laksa189.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.kristakingmath.com/blog/definition-of-the-laplace-transform\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.kristakingmath.com/blog/definition-of-the-laplace-transform\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/www.kristakingmath.com\/blog\/definition-of-the-laplace-transform","crawl_time":1775838171,"first_indexed_time":1596851166,"http_code":200,"src_unparsed":"com,kristakingmath!www,\/blog\/definition-of-the-laplace-transform s443","src_root_hash":"11392732973428258389","history_drop_reason":null,"meta_title":"Using the definition of the Laplace transform — Krista King Math | Online math help","meta_descriptions":["To find the Laplace transform of L using the definition of the Laplace transform, we’ll need to multiply f(t) by e^(-st), then integrate that product on the interval [0,infinity). This is the definition of the Laplace transform, such that the result is the Laplace transform of f(t), which we write as F(s)."],"attrs_boilerpipe_text":"Formula for the definition of the Laplace transform\nTo solve a Laplace transform in the form ???L\\left\\{f(t)\\right\\}??? using the definition of the Laplace transform, you’ll need to solve the equation\n???F(s)=\\int^\\infty_0{e}^{-st}f(t)\\ dt???\nwhere ???F(s)??? is the Laplace transform of the function ???f(t)???, ???s??? is a constant, and ???f(t)??? is the given function.\nHi! I'm krista.\nI create online courses to help you rock your math class.\nRead more.\nApplying the definition of the Laplace transform in order to transform a differential equation\nTake the course\nWant to learn more about Differential Equations? I have a step-by-step course for that. :)\nLaplace transform for an exponential function\nExample\nUse the definition to find the Laplace transform of the function.\n???f(t)=e^{-2t}???\nPlugging the given function into the definition of the Laplace transform gives\n???F(s)=\\int^\\infty_0{e}^{-st}f(t)\\ dt???\n???F(s)=\\int^\\infty_0{e}^{-st}e^{-2t}\\ dt???\n???F(s)=\\int^\\infty_0{e}^{-st-2t}\\ dt???\n???F(s)=\\int^\\infty_0{e}^{(-s-2)t}\\ dt???\nTo Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation.\nSince ???s??? is a constant, we can integrate and say\n???F(s)=\\frac{1}{-s-2}e^{(-s-2)t}\\Big|^\\infty_0???\n???F(s)=\\lim_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)t}\\Big|^b_0???\nEvaluate over the interval.\n???F(s)=\\lim_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)b}-\\frac{1}{-s-2}e^{(-s-2)0}???\n???F(s)=\\lim_{b\\to\\infty}\\frac{1}{-s-2}e^{-(sb+2b)}-\\frac{1}{-s-2}e^0???\n???F(s)=\\frac{1}{-s-2}e^{-\\infty}-\\frac{1}{-s-2}e^0???\nSince ???e^{-\\infty}=0???,\n???F(s)=\\frac{1}{-s-2}(0)-\\frac{1}{-s-2}(1)???\n???F(s)=-\\frac{1}{-s-2}???\n???F(s)=\\frac{1}{s+2}???\nThis is the Laplace transform of ???f(t)=e^{-2t}???.\nGet access to the complete Differential Equations course\nLearn math\nAugust 6, 2020\nmath\n,\nlearn online\n,\nonline course\n,\nonline math\n,\ndifferential equations\n,\nlaplace transforms\n,\ndefinition of the laplace transform\n,\nlaplace definition\n,\ntransforming f(t)\n,\nF(s)","attrs_markdown":"[![Krista King Math \\| Online math help](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1537392072530-1REILIR8TYPC00IS2EAU\/KKM_primary-white.png)](https:\/\/www.kristakingmath.com\/)\n\n[About](https:\/\/www.kristakingmath.com\/about-krista)   [Pricing](https:\/\/www.kristakingmath.com\/pricing)   [Login](https:\/\/courses.kristakingmath.com\/auth\/sign-in\/)\n\n[Subscribe Risk Free](https:\/\/courses.kristakingmath.com\/subscribe\/)\n\n[About](https:\/\/www.kristakingmath.com\/about-krista)[Pricing](https:\/\/www.kristakingmath.com\/pricing)[Login](https:\/\/courses.kristakingmath.com\/auth\/sign-in\/)\n\n[![Krista King Math \\| Online math help](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1537392072530-1REILIR8TYPC00IS2EAU\/KKM_primary-white.png)](https:\/\/www.kristakingmath.com\/)\n\nStep-by-step math courses covering Pre-Algebra through Calculus 3.\n\n[Subscribe Risk Free](https:\/\/courses.kristakingmath.com\/subscribe\/)\n\n# Using the definition of the Laplace transform\n![Laplace transforms using the definition blog post.jpeg](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1554828311071-TQ46EAP29XB5OVNO1G1L\/Laplace+transforms+using+the+definition+blog+post.jpeg)\n\n## Formula for the definition of the Laplace transform\nTo solve a Laplace transform in the form ???L\\\\left\\\\{f(t)\\\\right\\\\}??? using the definition of the Laplace transform, you’ll need to solve the equation\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st}f(t)\\\\ dt???\n\nwhere ???F(s)??? is the Laplace transform of the function ???f(t)???, ???s??? is a constant, and ???f(t)??? is the given function.\n\n![Krista King Math.jpg](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1531240577906-UIAUA4KAUI0TW2GDQK0M\/Krista+King+Math.jpg)\n\nHi! I'm krista.\n\nI create online courses to help you rock your math class. [Read more.](http:\/\/kristakingmath.com\/)\n\n## Applying the definition of the Laplace transform in order to transform a differential equation\n***\n\n![Krista King Math Signup.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1603386579776-6OCOSA9YN7OYTR6VTSWF\/Krista+King+Math+Signup.png)\n\n![Differential Equations course.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1550464378472-H2A166PJ27LF62ZHNKNW\/Differential+Equations+course.png)\n\n## Take the course\n### Want to learn more about Differential Equations? I have a step-by-step course for that. :)\n\n[Learn More](https:\/\/courses.kristakingmath.com\/library\/differential-equations-174134\/about\/)\n***\n## Laplace transform for an exponential function\n**Example**\n\nUse the definition to find the Laplace transform of the function.\n\n???f(t)=e^{-2t}???\n\nPlugging the given function into the definition of the Laplace transform gives\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st}f(t)\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st}e^{-2t}\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st-2t}\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{(-s-2)t}\\\\ dt???\n\n![Laplace transforms using the definition for Differential Equations.jpg](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1554828474394-4JMBC8CQCD740FS80U3X\/Laplace+transforms+using+the+definition+for+Differential+Equations.jpg)\n\nTo Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation.\n\nSince ???s??? is a constant, we can integrate and say\n\n???F(s)=\\\\frac{1}{-s-2}e^{(-s-2)t}\\\\Big\\|^\\\\infty\\_0???\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{(-s-2)t}\\\\Big\\|^b\\_0???\n\nEvaluate over the interval.\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{(-s-2)b}-\\\\frac{1}{-s-2}e^{(-s-2)0}???\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{-(sb+2b)}-\\\\frac{1}{-s-2}e^0???\n\n???F(s)=\\\\frac{1}{-s-2}e^{-\\\\infty}-\\\\frac{1}{-s-2}e^0???\n\nSince ???e^{-\\\\infty}=0???,\n\n???F(s)=\\\\frac{1}{-s-2}(0)-\\\\frac{1}{-s-2}(1)???\n\n???F(s)=-\\\\frac{1}{-s-2}???\n\n???F(s)=\\\\frac{1}{s+2}???\n\nThis is the Laplace transform of ???f(t)=e^{-2t}???.\n\n![Krista King.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1531243413505-L2JSZEAV47A16TR6BS72\/Krista+King.png)\n***\n### Get access to the complete Differential Equations course\n[Get started](https:\/\/courses.kristakingmath.com\/library\/differential-equations-174134\/about\/)\n***\n\n[Learn math](https:\/\/www.kristakingmath.com\/blog\/category\/Learn+math)[Krista King](https:\/\/www.kristakingmath.com\/blog?author=533db07be4b0f0bdc6e145a3)\n\nAugust 6, 2020\n\n[math](https:\/\/www.kristakingmath.com\/blog\/tag\/math), [learn online](https:\/\/www.kristakingmath.com\/blog\/tag\/learn+online), [online course](https:\/\/www.kristakingmath.com\/blog\/tag\/online+course), [online math](https:\/\/www.kristakingmath.com\/blog\/tag\/online+math), [differential equations](https:\/\/www.kristakingmath.com\/blog\/tag\/differential+equations), [laplace transforms](https:\/\/www.kristakingmath.com\/blog\/tag\/laplace+transforms), [definition of the laplace transform](https:\/\/www.kristakingmath.com\/blog\/tag\/definition+of+the+laplace+transform), [laplace definition](https:\/\/www.kristakingmath.com\/blog\/tag\/laplace+definition), [transforming 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Math.jpg](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1531240577906-UIAUA4KAUI0TW2GDQK0M\/Krista+King+Math.jpg)\n\nHi! I'm krista.\n\nI create online courses to help you rock your math class. [Read more.](http:\/\/kristakingmath.com\/)\n\nApplying the definition of the Laplace transform in order to transform a differential equation\n\n![Krista King Math Signup.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1603386579776-6OCOSA9YN7OYTR6VTSWF\/Krista+King+Math+Signup.png)\n\n![Differential Equations course.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1550464378472-H2A166PJ27LF62ZHNKNW\/Differential+Equations+course.png)\n\nTake the courseWant to learn more about Differential Equations? I have a step-by-step course for that. :)\n\n## Laplace transform for an exponential function\n**Example**\n\nUse the definition to find the Laplace transform of the function.\n\n???f(t)=e^{-2t}???\n\nPlugging the given function into the definition of the Laplace transform gives\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st}f(t)\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st}e^{-2t}\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{-st-2t}\\\\ dt???\n\n???F(s)=\\\\int^\\\\infty\\_0{e}^{(-s-2)t}\\\\ dt???\n\n![Laplace transforms using the definition for Differential Equations.jpg](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1554828474394-4JMBC8CQCD740FS80U3X\/Laplace+transforms+using+the+definition+for+Differential+Equations.jpg)\n\nTo Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation.\n\nSince ???s??? is a constant, we can integrate and say\n\n???F(s)=\\\\frac{1}{-s-2}e^{(-s-2)t}\\\\Big\\|^\\\\infty\\_0???\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{(-s-2)t}\\\\Big\\|^b\\_0???\n\nEvaluate over the interval.\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{(-s-2)b}-\\\\frac{1}{-s-2}e^{(-s-2)0}???\n\n???F(s)=\\\\lim\\_{b\\\\to\\\\infty}\\\\frac{1}{-s-2}e^{-(sb+2b)}-\\\\frac{1}{-s-2}e^0???\n\n???F(s)=\\\\frac{1}{-s-2}e^{-\\\\infty}-\\\\frac{1}{-s-2}e^0???\n\nSince ???e^{-\\\\infty}=0???,\n\n???F(s)=\\\\frac{1}{-s-2}(0)-\\\\frac{1}{-s-2}(1)???\n\n???F(s)=-\\\\frac{1}{-s-2}???\n\n???F(s)=\\\\frac{1}{s+2}???\n\nThis is the Laplace transform of ???f(t)=e^{-2t}???.\n\n![Krista King.png](https:\/\/images.squarespace-cdn.com\/content\/v1\/533db07de4b0d9f7ba7f1e77\/1531243413505-L2JSZEAV47A16TR6BS72\/Krista+King.png)\n\nGet access to the complete Differential Equations course\n\n[Learn math](https:\/\/www.kristakingmath.com\/blog\/category\/Learn+math)August 6, 2020[math](https:\/\/www.kristakingmath.com\/blog\/tag\/math), [learn online](https:\/\/www.kristakingmath.com\/blog\/tag\/learn+online), [online course](https:\/\/www.kristakingmath.com\/blog\/tag\/online+course), [online math](https:\/\/www.kristakingmath.com\/blog\/tag\/online+math), [differential equations](https:\/\/www.kristakingmath.com\/blog\/tag\/differential+equations), [laplace transforms](https:\/\/www.kristakingmath.com\/blog\/tag\/laplace+transforms), [definition of the laplace transform](https:\/\/www.kristakingmath.com\/blog\/tag\/definition+of+the+laplace+transform), [laplace definition](https:\/\/www.kristakingmath.com\/blog\/tag\/laplace+definition), [transforming f(t)](https:\/\/www.kristakingmath.com\/blog\/tag\/transforming+f%28t%29), [F(s)](https:\/\/www.kristakingmath.com\/blog\/tag\/F%28s%29)","meta_canonical":null}

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Meta Description	To find the Laplace transform of L using the definition of the Laplace transform, we’ll need to multiply f(t) by e^(-st), then integrate that product on the interval [0,infinity). This is the definition of the Laplace transform, such that the result is the Laplace transform of f(t), which we write as F(s).
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Boilerpipe Text	Formula for the definition of the Laplace transform To solve a Laplace transform in the form ???L\left\{f(t)\right\}??? using the definition of the Laplace transform, you’ll need to solve the equation ???F(s)=\int^\infty_0{e}^{-st}f(t)\ dt??? where ???F(s)??? is the Laplace transform of the function ???f(t)???, ???s??? is a constant, and ???f(t)??? is the given function. Hi! I'm krista. I create online courses to help you rock your math class. Read more. Applying the definition of the Laplace transform in order to transform a differential equation Take the course Want to learn more about Differential Equations? I have a step-by-step course for that. :) Laplace transform for an exponential function Example Use the definition to find the Laplace transform of the function. ???f(t)=e^{-2t}??? Plugging the given function into the definition of the Laplace transform gives ???F(s)=\int^\infty_0{e}^{-st}f(t)\ dt??? ???F(s)=\int^\infty_0{e}^{-st}e^{-2t}\ dt??? ???F(s)=\int^\infty_0{e}^{-st-2t}\ dt??? ???F(s)=\int^\infty_0{e}^{(-s-2)t}\ dt??? To Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation. Since ???s??? is a constant, we can integrate and say ???F(s)=\frac{1}{-s-2}e^{(-s-2)t}\Big\|^\infty_0??? ???F(s)=\lim_{b\to\infty}\frac{1}{-s-2}e^{(-s-2)t}\Big\|^b_0??? Evaluate over the interval. ???F(s)=\lim_{b\to\infty}\frac{1}{-s-2}e^{(-s-2)b}-\frac{1}{-s-2}e^{(-s-2)0}??? ???F(s)=\lim_{b\to\infty}\frac{1}{-s-2}e^{-(sb+2b)}-\frac{1}{-s-2}e^0??? ???F(s)=\frac{1}{-s-2}e^{-\infty}-\frac{1}{-s-2}e^0??? Since ???e^{-\infty}=0???, ???F(s)=\frac{1}{-s-2}(0)-\frac{1}{-s-2}(1)??? ???F(s)=-\frac{1}{-s-2}??? ???F(s)=\frac{1}{s+2}??? This is the Laplace transform of ???f(t)=e^{-2t}???. Get access to the complete Differential Equations course Learn math August 6, 2020 math , learn online , online course , online math , differential equations , laplace transforms , definition of the laplace transform , laplace definition , transforming f(t) , F(s)
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[Subscribe Risk Free](https://courses.kristakingmath.com/subscribe/) # Using the definition of the Laplace transform ![Laplace transforms using the definition blog post.jpeg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1554828311071-TQ46EAP29XB5OVNO1G1L/Laplace+transforms+using+the+definition+blog+post.jpeg) ## Formula for the definition of the Laplace transform To solve a Laplace transform in the form ???L\\left\\{f(t)\\right\\}??? using the definition of the Laplace transform, you’ll need to solve the equation ???F(s)=\\int^\\infty\_0{e}^{-st}f(t)\\ dt??? where ???F(s)??? is the Laplace transform of the function ???f(t)???, ???s??? is a constant, and ???f(t)??? is the given function. ![Krista King Math.jpg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1531240577906-UIAUA4KAUI0TW2GDQK0M/Krista+King+Math.jpg) Hi! I'm krista. I create online courses to help you rock your math class. [Read more.](http://kristakingmath.com/) ## Applying the definition of the Laplace transform in order to transform a differential equation * ![Krista King Math Signup.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1603386579776-6OCOSA9YN7OYTR6VTSWF/Krista+King+Math+Signup.png) ![Differential Equations course.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1550464378472-H2A166PJ27LF62ZHNKNW/Differential+Equations+course.png) ## Take the course ### Want to learn more about Differential Equations? I have a step-by-step course for that. :) [Learn More](https://courses.kristakingmath.com/library/differential-equations-174134/about/) * ## Laplace transform for an exponential function Example Use the definition to find the Laplace transform of the function. ???f(t)=e^{-2t}??? Plugging the given function into the definition of the Laplace transform gives ???F(s)=\\int^\\infty\_0{e}^{-st}f(t)\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{-st}e^{-2t}\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{-st-2t}\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{(-s-2)t}\\ dt??? ![Laplace transforms using the definition for Differential Equations.jpg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1554828474394-4JMBC8CQCD740FS80U3X/Laplace+transforms+using+the+definition+for+Differential+Equations.jpg) To Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation. Since ???s??? is a constant, we can integrate and say ???F(s)=\\frac{1}{-s-2}e^{(-s-2)t}\\Big\\|^\\infty\_0??? ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)t}\\Big\\|^b\_0??? Evaluate over the interval. ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)b}-\\frac{1}{-s-2}e^{(-s-2)0}??? ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{-(sb+2b)}-\\frac{1}{-s-2}e^0??? ???F(s)=\\frac{1}{-s-2}e^{-\\infty}-\\frac{1}{-s-2}e^0??? Since ???e^{-\\infty}=0???, ???F(s)=\\frac{1}{-s-2}(0)-\\frac{1}{-s-2}(1)??? ???F(s)=-\\frac{1}{-s-2}??? ???F(s)=\\frac{1}{s+2}??? This is the Laplace transform of ???f(t)=e^{-2t}???. ![Krista King.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1531243413505-L2JSZEAV47A16TR6BS72/Krista+King.png) * ### Get access to the complete Differential Equations course [Get started](https://courses.kristakingmath.com/library/differential-equations-174134/about/) * [Learn math](https://www.kristakingmath.com/blog/category/Learn+math)[Krista King](https://www.kristakingmath.com/blog?author=533db07be4b0f0bdc6e145a3) August 6, 2020 [math](https://www.kristakingmath.com/blog/tag/math), [learn online](https://www.kristakingmath.com/blog/tag/learn+online), [online course](https://www.kristakingmath.com/blog/tag/online+course), [online math](https://www.kristakingmath.com/blog/tag/online+math), [differential equations](https://www.kristakingmath.com/blog/tag/differential+equations), [laplace transforms](https://www.kristakingmath.com/blog/tag/laplace+transforms), [definition of the laplace transform](https://www.kristakingmath.com/blog/tag/definition+of+the+laplace+transform), [laplace definition](https://www.kristakingmath.com/blog/tag/laplace+definition), [transforming f(t)](https://www.kristakingmath.com/blog/tag/transforming+f%28t%29), [F(s)](https://www.kristakingmath.com/blog/tag/F%28s%29) [Facebook0](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Fwww.kristakingmath.com%2Fblog%2Fdefinition-of-the-laplace-transform) [Twitter](https://twitter.com/intent/tweet?url=https%3A%2F%2Fwww.kristakingmath.com%2Fblog%2Fdefinition-of-the-laplace-transform&text=To+find+the+Laplace+transform+of+L+using+the+definition+of+the+Laplace+transform%2C+we%E2%80%99ll+need+to+...) 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Readable Markdown	![Laplace transforms using the definition blog post.jpeg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1554828311071-TQ46EAP29XB5OVNO1G1L/Laplace+transforms+using+the+definition+blog+post.jpeg) ## Formula for the definition of the Laplace transform To solve a Laplace transform in the form ???L\\left\\{f(t)\\right\\}??? using the definition of the Laplace transform, you’ll need to solve the equation ???F(s)=\\int^\\infty\_0{e}^{-st}f(t)\\ dt??? where ???F(s)??? is the Laplace transform of the function ???f(t)???, ???s??? is a constant, and ???f(t)??? is the given function. ![Krista King Math.jpg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1531240577906-UIAUA4KAUI0TW2GDQK0M/Krista+King+Math.jpg) Hi! I'm krista. I create online courses to help you rock your math class. [Read more.](http://kristakingmath.com/) Applying the definition of the Laplace transform in order to transform a differential equation ![Krista King Math Signup.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1603386579776-6OCOSA9YN7OYTR6VTSWF/Krista+King+Math+Signup.png) ![Differential Equations course.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1550464378472-H2A166PJ27LF62ZHNKNW/Differential+Equations+course.png) Take the courseWant to learn more about Differential Equations? I have a step-by-step course for that. :) ## Laplace transform for an exponential function Example Use the definition to find the Laplace transform of the function. ???f(t)=e^{-2t}??? Plugging the given function into the definition of the Laplace transform gives ???F(s)=\\int^\\infty\_0{e}^{-st}f(t)\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{-st}e^{-2t}\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{-st-2t}\\ dt??? ???F(s)=\\int^\\infty\_0{e}^{(-s-2)t}\\ dt??? ![Laplace transforms using the definition for Differential Equations.jpg](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1554828474394-4JMBC8CQCD740FS80U3X/Laplace+transforms+using+the+definition+for+Differential+Equations.jpg) To Find the Laplace transform of a function using the definition of the Laplace transform, we’ll need to solve an integral equation. Since ???s??? is a constant, we can integrate and say ???F(s)=\\frac{1}{-s-2}e^{(-s-2)t}\\Big\\|^\\infty\_0??? ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)t}\\Big\\|^b\_0??? Evaluate over the interval. ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{(-s-2)b}-\\frac{1}{-s-2}e^{(-s-2)0}??? ???F(s)=\\lim\_{b\\to\\infty}\\frac{1}{-s-2}e^{-(sb+2b)}-\\frac{1}{-s-2}e^0??? ???F(s)=\\frac{1}{-s-2}e^{-\\infty}-\\frac{1}{-s-2}e^0??? Since ???e^{-\\infty}=0???, ???F(s)=\\frac{1}{-s-2}(0)-\\frac{1}{-s-2}(1)??? ???F(s)=-\\frac{1}{-s-2}??? ???F(s)=\\frac{1}{s+2}??? This is the Laplace transform of ???f(t)=e^{-2t}???. ![Krista King.png](https://images.squarespace-cdn.com/content/v1/533db07de4b0d9f7ba7f1e77/1531243413505-L2JSZEAV47A16TR6BS72/Krista+King.png) Get access to the complete Differential Equations course [Learn math](https://www.kristakingmath.com/blog/category/Learn+math)August 6, 2020[math](https://www.kristakingmath.com/blog/tag/math), [learn online](https://www.kristakingmath.com/blog/tag/learn+online), [online course](https://www.kristakingmath.com/blog/tag/online+course), [online math](https://www.kristakingmath.com/blog/tag/online+math), [differential equations](https://www.kristakingmath.com/blog/tag/differential+equations), [laplace transforms](https://www.kristakingmath.com/blog/tag/laplace+transforms), [definition of the laplace transform](https://www.kristakingmath.com/blog/tag/definition+of+the+laplace+transform), [laplace definition](https://www.kristakingmath.com/blog/tag/laplace+definition), [transforming f(t)](https://www.kristakingmath.com/blog/tag/transforming+f%28t%29), [F(s)](https://www.kristakingmath.com/blog/tag/F%28s%29)
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