DEFINITE INTEGRALS CONTAINING EXPONENTIAL FUNCTIONS

1.

$$\int_{0}^{\infty}e^{-ax}\cos bx dx=\displaystyle \frac{a}{a^2+b^2}$$

2.

$$\int_{0}^{\infty}e^{-ax}\sin bx dx=\displaystyle \frac{b}{a^2+b^2}$$

3.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-ax}\sin bx}{x}dx=\tan^{-1}\displaystyle \frac{b}{a}$$

4.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-ax}-e^{-bx}}{x}dx=\ln\displaystyle \frac{b}{a}$$

5.

$$\int_{0}^{\infty}e^{-ax^2}dx=\displaystyle \frac{1}{2}\displaystyle \sqrt{\displaystyle \frac{\pi}{a}}$$

6.

$$\int_{0}^{\infty}e^{-ax^2}\cos bx dx=\displaystyle \frac{1}{2}\displaystyle \sqrt{\displaystyle \frac{\pi}{a}}e^{-b^2/4a}$$

7.

$$\int_{0}^{\infty}e^{-(ax^2+bx+c)}dx=\displaystyle \frac{1}{2}\displaystyle \sqrt{\displaystyle \frac{\pi}{a}}e^{(b^2-4ac)/4a}$$

8.

$$\int_{-\infty}^{\infty}e^{-(ax^2+bx+c)}dx=\displaystyle \sqrt{\displaystyle \frac{\pi}{a}}e^{(b^2-4ac)/4a}$$

9.

$$\int_{0}^{\infty}x^n e^{-ax}dx=\displaystyle \frac{\Gamma(n+1)}{a^n+1}$$

10.

$$\int_{0}^{\infty}x^m e^{-ax^2}dx=\displaystyle \frac{\Gamma[(m+1)/2]}{2a^{(m+1)/2}}$$

11.

$$\int_{0}^{\infty}e^{-(ax^2+b/x^2)}dx=\displaystyle \frac{1}{2}\displaystyle \sqrt{\displaystyle \frac{\pi}{a}}e^{-2\displaystyle \sqrt{ab}}$$

12.

$$\int_{0}^{\infty}\displaystyle \frac{xdx}{e^x-1}=\displaystyle \fr... ...3^2}+\displaystyle \frac{1}{4^2}+\cdot\cdot\cdot =\displaystyle \frac{\pi^2}{6}$$

13.

$$\int_{0}^{\infty}\displaystyle \frac{x^{n-1}}{e^x-1}dx=\Gamma(n+1)... ...\displaystyle \frac{1}{2^n}+\displaystyle \frac{1}{3^n}+\cdot\cdot\cdot \right)$$

14.

$$\int_{0}^{\infty}\displaystyle \frac{xdx}{e^x+1}=\displaystyle \fr... ...2}-\displaystyle \frac{1}{4^2}+ \cdot\cdot\cdot =\displaystyle \frac{\pi^2}{12}$$

15.

$$\int_{0}^{\infty}\displaystyle \frac{x^{n-1}}{e^x+1}dx=\Gamma(n+1)... ...\displaystyle \frac{1}{2^n}+\displaystyle \frac{1}{3^n}-\cdot\cdot\cdot \right)$$

16.

$$\int_{0}^{\infty}\displaystyle \frac{\sin mx}{e^{2\pi x}-1}dx=\displaystyle \frac{1}{4}\coth\displaystyle \frac{m}{2}-\displaystyle \frac{1}{2m}$$

17.

$$\int_{0}^{\infty}\left( \displaystyle \frac{1}{1+x}-e^{-x}\right)\displaystyle \frac{dx}{x}=\gamma$$

where the constant $\gamma$ is the Euler's constant.

18.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-x^2}-e^{-x}}{x}dx=\displaystyle \frac{1}{2}\gamma$$

where the constant $\gamma$ is the Euler's constant.

19.

$$\int_{0}^{\infty}\left( \displaystyle \frac{1}{e^x-1}-\displaystyle \frac{e^{-x}}{x}\right)dx=\gamma$$

where the constant $\gamma$ is the Euler's constant.

20.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-ax}-e^{-bx}}{x\sec px}dx=\displaystyle \frac{1}{2}\ln\left(\displaystyle \frac{b^2+p^2}{a^2+p^2}\right)$$

21.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-ax}-e^{-bx}}{x\csc px}dx=\tan^{-1}\displaystyle \frac{b}{p}-\tan^{-1}\displaystyle \frac{a}{p}$$

22.

$$\int_{0}^{\infty}\displaystyle \frac{e^{-ax}(1-\cos x)}{x^2}dx=\cot^{-1}a-\displaystyle \frac{a}{2}\ln(a^2+1)$$

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