INTEGRALS CONTAINING e^ax

1.

$$\int e^{ax} dx =\displaystyle \frac{e^{ax}}{a}$$

2.

$$\int xe^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x-\displaystyle \frac{1}{a}\right)$$

3.

$$\int x^2 e^{ax}dx=\displaystyle \frac{e^{ax}}{a}\left(x^2-\displaystyle \frac{2x}{a}+\displaystyle \frac{2}{a^2}\right)$$

4.

$\begin{array}{lcl} \displaystyle\int x^n e^{ax}dx&=& \displaystyle \frac{x^n e^... ...2}}{a^2}-\cdot\cdot\cdot \displaystyle \frac{(-1)^n n!}{a^n}\right) \end{array}$

5.

$$\int\displaystyle \frac{e^{ax}}{x}dx=\ln x+\displaystyle \frac{ax}... ...\frac{(ax)^2}{2\cdot 2!}+\displaystyle \frac{(ax)^3}{3\cdot 3!}+\cdot\cdot\cdot$$

6.

$$\int\displaystyle \frac{e^{ax}}{x^n}dx=\displaystyle \frac{-e^{ax}... ...)x^{n-1}}+\displaystyle \frac{a}{n-1}\int\displaystyle \frac{e^{ax}}{x^{n-1}}dx$$

7.

$$\int\displaystyle \frac{dx}{p+qe^{ax}}=\displaystyle \frac{x}{p}-\displaystyle \frac{1}{ap}\ln (p+qe^{ax})$$

8.

$$\int\displaystyle \frac{dx}{(p+qe^{ax})^2}=\displaystyle \frac{x}{... ...displaystyle \frac{1}{ap(p+qe^{ax})}-\displaystyle \frac{1}{ap^2}\ln(p+qe^{ax})$$

9.

$$\int\displaystyle \frac{dx}{pe^{ax}+qe^{-ax}}=\left\{ \begin{array... ...style \sqrt{-q/p}}{e^{ax}+\displaystyle \sqrt{-q/p}}\right) \end{array}\right.$$

10.

$$\int e^{ax}\sin bx dx=\displaystyle \frac{e^{ax}(a\sin bx -b\cos bx)}{a^2+b^2}$$

11.

$$\int e^{ax}\cos bx dx=e^{ax}\displaystyle \frac{(a\cos bx+b\sin bx)}{a^2+b^2}$$

12.

$$\int xe^{ax}\sin bx dx=\displaystyle \frac{xe^{ax}(a\sin bx -b\cos... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\sin bx-2ab\cos bx\right\}}{(a^2+b^2)^2}$$

13.

$$\int xe^{ax}\cos bx dx=\displaystyle \frac{xe^{ax}(a\cos bx +b\sin... ...splaystyle \frac{e^{ax}\left\{(a^2-b^2)\cos bx+2ab\sin bx\right\}}{(a^2+b^2)^2}$$

14.

$$\int e^{ax}\ln xdx=\displaystyle \frac{e^{ax}\ln x}{a}-\displaystyle \frac{1}{a}\int\displaystyle \frac{e^{ax}}{x}dx$$

15.

$$\int e^{ax}\sin^n bxdx=\displaystyle \frac{e^{ax}\sin^{n-1}bx}{a^2... ...\cos bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\sin^{n-2}bx dx$$

16.

$$\int e^{ax}\cos^n bxdx=\displaystyle \frac{e^{ax}\cos^{n-1}bx}{a^2... ...\sin bx) + \displaystyle \frac{n(n-1)b^2}{a^2+n^2b^2}\int e^{ax}\cos^{n-2}bx dx$$

[Tables]

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