Improper integration involving complex analytic arguments

I am trying to evaluate the following:

$\displaystyle \int_{0}^{\infty} \frac{1}{1+x^a}dx$, where $a>1$ and $a \in \mathbb{R}$

Any help will be much appreciated.

Answer

Use the change of variables $1+x^\alpha=\frac{1}{t}$ to cast the integral in terms of the beta function

$$ \frac{1}{\alpha}\int_{0}^{1}t^{-1/\alpha}(1-t)^{1/\alpha-1}= \frac{1}{\alpha}\Gamma\left(\frac{1}{\alpha}\right)\Gamma\left(1-\frac{1}{\alpha}\right) $$