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|<math> \int F { f ( x ) } d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^' ( x )} d u \qquad u = f ( x ) </math> | |<math> \int F { f ( x ) } d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^' ( x )} d u \qquad u = f ( x ) </math> | ||
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| + | |<math> \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 </math> | ||
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| + | |<math> \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ or \ln {-u} \ ( if \ u < 0 ) = \ln \left | u \right | </math> | ||
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| + | |<math> \int e^u d u = e^u </math> | ||
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| + | |<math> \int a^u d u = \int e^{u \ln a} d u = \frac{e^{u \ln a}}{\ln a} = \frac{a^u}{\ln a} \qquad a > 0 \ and \ a \neq 1</math> | ||
Revision as of 13:53, 13 November 2010
| Table of Infinite Integrals | |
|---|---|
| General Rules | |
| $ \int a d x = a x $ | |
| $ \int a f ( x ) d x = a \int f ( x ) d x $ | |
| $ \int ( u \pm v \pm w \pm \cdot \cdot \cdot ) d x = \int u d x \pm \int v d x \pm \int w d x \pm \cdot \cdot \cdot $ | |
| $ \int u d v = u v - \int v d u $ | |
| $ \int f ( a x ) d x = \frac{1}{a} \int f ( u ) d u $ | |
| $ \int F { f ( x ) } d x = \int F ( u ) \frac{dx}{du} d u = \int \frac{F ( u )}{f^' ( x )} d u \qquad u = f ( x ) $ | |
| $ \int u^n d u = \frac{u^{n+1}}{n+1} \qquad n \neq -1 $ | |
| $ \int \frac{d u}{u} = \ln u \ ( if \ u > 0 ) \ or \ln {-u} \ ( if \ u < 0 ) = \ln \left | u \right | $ | |
| $ \int e^u d u = e^u $ | |
| $ \int a^u d u = \int e^{u \ln a} d u = \frac{e^{u \ln a}}{\ln a} = \frac{a^u}{\ln a} \qquad a > 0 \ and \ a \neq 1 $ | |
