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## Definite Integrals of Rational Functions

$$\int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}$$   $$\int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1$$   $$\int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n$$   $$\int^a_0 \frac{dx}{\sqrt{a^2 – x^2}} = \frac{\pi}{2}$$   $$\int^a_0 \sqrt{a^2 – x^2}\,dx = \frac{\pi\,a^2}{4}$$   […]