Definite Integrals with Rational and Irrational Expressions

1.
$\displaystyle \int_{0}^{\infty} \frac{dx}{\displaystyle x^2 + a^2} = \displaystyle\frac{\pi}{2a}$

2.
$\displaystyle \int_{0}^{\infty} \frac{\displaystyle x^{p-1}dx}{1+x} = \displaystyle \frac{\pi}{\sin(p\pi)}$,
for 0 < p <1

3.
$\displaystyle \int_{0}^{\infty} \frac{x^m}{\displaystyle x^n + a^n} = \displaystyle \frac{\pi a^{m+1-n}}{n \sin\Big[(m+1)\pi/n\Big]}$,
for 0 < m+1 < n

4.
$\displaystyle \int_{0}^{\infty} \frac{x^m}{\displaystyle1 + 2x \cos(b) + x^2} = \displaystyle \frac{\pi}{\sin(m\pi)} \displaystyle \frac{\sin(mb)}{\sin(b)}$

5.
$\displaystyle \int_{0}^{a} \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}$

6.
$\displaystyle \int_{0}^{a} \sqrt{a^2-x^2} dx = \frac{\pi a^2}{4}$

7.
$\displaystyle \int_{0}^{a} x^m (a^n-x^n)^pdx = \frac{a^{m+1+np} \Gamma\Big[(m+1)/n\Big] \Gamma(p+1)}{n \Gamma\Big[(m+1)/(n+p+1)\Big]}$

8.
$\displaystyle \int_{0}^{\infty} \frac{x^m}{(a^n+x^n)^p}dx = \frac{(-1)^{p-1} \p...
...g[(m+1)/n\Big]}{n \sin\Big[(m+1)\pi/n\Big] (p-1)!\Gamma\Big[(m+1)/(n-p+1)\Big]}$,
for 0 < m+1< n p

[Tables]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour