“우리 대부분은 초라한 옷차림과 엉터리 가구들을 부끄럽게 여기지만, 초라한 생각과 엉터리 철학을 부끄럽게 여길 줄 알아야 한다.” – Albert Einstein

19900101
calculus integrals

https://www.eeweb.com/tools/calculus-integrals-sheet

Integrals

Definition of an Integral

The integral is a mathematical analysis applied to a function that results in the area bounded by the graph of the function, x axis, and limits of the integral. Integrals can be referred to as anti-derivatives, because the derivative of the integral of a function is equal to the function.

Properties

latex!encoded:base64,XGludF97YX1ee2J9IGYoeCkgXHBtIGcoeCkgZHggPSBcaW50X3thfV57Yn0gZih4KSBkeCBccG0gXGludF97YX1ee2J9IGcoeCkgZHg=
latex!encoded:base64,XGludF97YX1ee2F9Zih4KWR4ID0gMA==
latex!encoded:base64,XGludF97YX1ee2J9Zih4KWR4ID0gLVxpbnRfe2J9XnthfWYoeClkeA==
latex!encoded:base64,XGludF97YX1ee2J9Y2YoeClkeCA9IGNcaW50X3thfV57Yn1mKHgpZHg=

Common Integrals

latex!encoded:base64,XGludCBrZHggPSBreCArIGM=
latex!encoded:base64,XGludCB4XntufWR4ID0gXGZyYWN7MX17bisxfXhee24rMX0rYyxuXG5lcSAtMQ==
latex!encoded:base64,XGludCB4XnstMX1keCA9IFxmcmFjezF9e3h9ZHg9bG58eHwrYw==
latex!encoded:base64,XGludCBcZnJhY3sxfXtheCtifWR4PVxmcmFjezF9e2F9bG58YXgrYnwrYw==
latex!encoded:base64,XGludCBsbih4KWR4PXhsbih4KS14K2M=
latex!encoded:base64,XGludCBlXnt4fWR4PWVee3h9K2M=
latex!encoded:base64,XGludCBjb3N4ZHg9c2lueCtj
latex!encoded:base64,XGludCBzaW54ZHg9LWNvc3grYw==
latex!encoded:base64,XGludCBcc2VjXnsyfXhkeD1cdGFuIHgrYw==
latex!encoded:base64,XGludFxzZWMgeCBcdGFuIHggZHg9XHNlYyB4K2M=
latex!encoded:base64,XGludCBcY3NjIHggXGNvdCB4ZHg9LVxjc2MgeCtj
latex!encoded:base64,XGludCBcY3NjXnsyfXhkeD0tXGNvdCB4K2M=
latex!encoded:base64,XGludCBcdGFuIHhkeD1sbnxcc2VjIHh8K2M=
latex!encoded:base64,XGludCBcc2VjIHhkeD1sbnxcc2VjIHgrXHRhbiB4fCtj
latex!encoded:base64,XGxhcmdlIFxpbnQgXGZyYWN7MX17YV57Mn0reF57Mn19ZHg9XGZyYWN7MX17YX1cdGFuXnstMX1cbGVmdCAoXGZyYWN7eH17YX0gXHJpZ2h0ICkrYw==
latex!encoded:base64,XGludCBcZnJhY3sxfXtcc3FydHthXnsyfS14XnsyfX19ZHg9XHNpbiBeey0xfVxsZWZ0ICggXGZyYWN7eH17YX0gXHJpZ2h0ICkrYw==

Integration by Substitution

latex!encoded:base64,XGxhcmdlIFxpbnRfe2F9XntifWZcbGVmdCAoIGcoeCkgXHJpZ2h0IClnJyh4KWR4PVxpbnRfe2coYSl9XntnKGIpfWYodSlkdQ==
latex!encoded:base64,dT1neA==
latex!encoded:base64,ZHU9ZycoeClkeA==

where     and  

Integration by Parts

latex!encoded:base64,XGludCB1IFwgZHYgPSB1diAtIFxpbnQgdiBcIGR1
latex!encoded:base64,diA9IFxpbnQgZHY=

where  

Integration by Trigonometric Substitution

Trigonometric identities can be use with integration substitution to simplify integrals. There are three common substitutions.

First Trigonometric Substitution
latex!encoded:base64,XGxhcmdlIFxzcXJ0e2FeezJ9LXheezJ9fQ==

To take advantage of the property

latex!encoded:base64,XGxhcmdlIDEtXHNpbiBeezJ9XHRoZXRhPVxjb3MgXnsyfVx0aGV0YQ==

Substitute

latex!encoded:base64,XGxhcmdlIHg9YVxzaW4gXHRoZXRh
latex!encoded:base64,XGxhcmdlIGR4PWFcY29zIFx0aGV0YSBkXHRoZXRh

After substitution

latex!encoded:base64,XGxhcmdlIFxzcXJ0e2FeezJ9LWFeezJ9XHNpbiBeezJ9XHRoZXRhfT1hXGNvcyBcdGhldGE=
Second Trigonometric Substitution
latex!encoded:base64,XGxhcmdlIFxzcXJ0e3heezJ9LWFeezJ9fQ==

To take advantage of the property

latex!encoded:base64,XGxhcmdlIFxzZWMgXnsyfVx0aGV0YSAtMT1cdGFuIF57Mn1cdGhldGE=

Substitute

latex!encoded:base64,XGxhcmdlIHg9YVxzZWMgXHRoZXRhCg==
latex!encoded:base64,XGxhcmdlIGR4PWEgXHNlYyBcdGhldGFcdGFuIFx0aGV0YSBkXHRoZXRh

After substitute

latex!encoded:base64,XGxhcmdlIFxzcXJ0e2FeezJ9XHNlYyBeezJ9XHRoZXRhIC1hXnsyfX09YVx0YW4gXHRoZXRh
Third Trigonometric Substitution
latex!encoded:base64,XGxhcmdlIFxzcXJ0e2FeezJ9K3heezJ9fQ==

To take advantage of the property

latex!encoded:base64,XGxhcmdlIDEgKyB0YW5eezJ9IFx0aGV0YSA9IHNlY157Mn1cdGhldGE=

Substitute

latex!encoded:base64,XGxhcmdlIHg9YVx0YW4gXHRoZXRh
latex!encoded:base64,XGxhcmdlIGR4PWFcc2VjIF57Mn1cdGhldGEgZCBcdGhldGE=

After substitute

latex!encoded:base64,XGxhcmdlIFxzcXJ0e2FeezJ9K2FeezJ9IFx0YW4gXnsyfVx0aGV0YX09YVxzZWMgXHRoZXRh

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