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|<math> \int x^n e^{ax}dx=\frac{x^n e^{ax}}{a}-\frac{n}{a} \int x^{n-1} e^{ax}dx = \frac {e^{ax}}{a} \left( x^n- \frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2}- \cdot \cdot \cdot \frac{(-1)^n n!}{a^n} \right ) \qquad \text{if n is a poaitive integer} </math> | |<math> \int x^n e^{ax}dx=\frac{x^n e^{ax}}{a}-\frac{n}{a} \int x^{n-1} e^{ax}dx = \frac {e^{ax}}{a} \left( x^n- \frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2}- \cdot \cdot \cdot \frac{(-1)^n n!}{a^n} \right ) \qquad \text{if n is a poaitive integer} </math> | ||
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| + | <math> \int \frac {e^{ax}}{x}dx=\ln {x} + \frac {ax}{1 \cdot 1!} + \frac {(ax)^2}{2 \cdot 2!} + \frac {(ax)^3}{3 \cdot 3!} + \cdot \cdot \cdot </math> | ||
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| + | <math> \frac {e^{ax}}{x^n}dx= \frac {-e^{ax}}{(n-1)x^{n-1}} + \frac {a}{n-1} \int \frac {e^{ax}}{x^{n-1}}dx </math> | ||
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| + | <math> \frac {dx}{p+qe^{ax}}=\frac {x}{p}-\frac {1}{ap} \ln {\left (p+qe^{ax}\right)} </math> | ||
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| + | <math> \frac {dx} {\left ( p+qe^{ax} \right) ^2}=\frac {x}{p^2}+\frac {1}{ap(p+qe^{ax})} -\frac{1}{ap^2}\ln {\left (p+qe^{ax}\right)} </math> | ||
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Revision as of 09:45, 24 November 2010
| Table of Infinite Integrals Continues | |
|---|---|
| 27 Integrals Component $ e^{ax} $ | |
| $ \int e^{ax}dx=\frac{e^{ax}}{a} $ | |
| $ \int x e^{ax}dx=\frac{e^{ax}}{a}\left(x-\frac{1}{a} \right) $ | |
| $ \int x^2 e^{ax}dx=\frac{e^{ax}}{a}\left(x^2-\frac{2x}{a}+\frac{2}{a^2}\right) $ | |
| $ \int x^n e^{ax}dx=\frac{x^n e^{ax}}{a}-\frac{n}{a} \int x^{n-1} e^{ax}dx = \frac {e^{ax}}{a} \left( x^n- \frac{nx^{n-1}}{a}+\frac{n(n-1)x^{n-2}}{a^2}- \cdot \cdot \cdot \frac{(-1)^n n!}{a^n} \right ) \qquad \text{if n is a poaitive integer} $ | |
