Definite Integrals with Rational and Irrational Expressions

1.

$$\int_{0}^{\infty} \frac{dx}{\displaystyle x^2 + a^2} = \displaystyle\frac{\pi}{2a}$$

2.

$$\int_{0}^{\infty} \frac{\displaystyle x^{p-1}dx}{1+x} = \displaystyle \frac{\pi}{\sin(p\pi)}$$,
for 0 < p <1

3.

$$\int_{0}^{\infty} \frac{x^m}{\displaystyle x^n + a^n} = \displaystyle \frac{\pi a^{m+1-n}}{n \sin\Big[(m+1)\pi/n\Big]}$$,
for 0 < m+1 < n

4.

$$\int_{0}^{\infty} \frac{x^m}{\displaystyle1 + 2x \cos(b) + x^2} = \displaystyle \frac{\pi}{\sin(m\pi)} \displaystyle \frac{\sin(mb)}{\sin(b)}$$

5.

$$\int_{0}^{a} \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}$$

6.

$$\int_{0}^{a} \sqrt{a^2-x^2} dx = \frac{\pi a^2}{4}$$

7.

$$\int_{0}^{a} x^m (a^n-x^n)^pdx = \frac{a^{m+1+np} \Gamma\Big[(m+1)/n\Big] \Gamma(p+1)}{n \Gamma\Big[(m+1)/(n+p+1)\Big]}$$

8.

$$\int_{0}^{\infty} \frac{x^m}{(a^n+x^n)^p}dx = \frac{(-1)^{p-1} \p... ...g[(m+1)/n\Big]}{n \sin\Big[(m+1)\pi/n\Big] (p-1)!\Gamma\Big[(m+1)/(n-p+1)\Big]}$$,
for 0 < m+1< n p

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