List of formulas related to definite integrals of rational functions
$$\int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a}$$
$$\int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1$$
$$ \int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n $$
$$ \int^a_0 \frac{dx}{\sqrt{a^2 - x^2}} = \frac{\pi}{2}$$
$$ \int^a_0 \sqrt{a^2 - x^2}\,dx = \frac{\pi\,a^2}{4}$$
$$ \int^a_0 x^m \left(a^n - x^n\right)^p\,dx = \frac{a^{m+1+np}~\Gamma[(m+1)/n]~\Gamma(p+1) }{n\,\Gamma[(m+1)/n + p +1]}$$