TITLE: Integration without complex analysis on rational-improper integral
QUESTION [5 upvotes]: Evaluate:
$$\int_{0}^{\infty} \frac{1}{x^6 + 1} \,\mathrm dx$$
Without the use of complex-analysis.
With complex analysis it is a very simple problem, how can this be done WITHOUT complex analysis?

REPLY [3 votes]: how can this be done WITHOUT complex analysis?

$\quad$ All integrals of the form $~\displaystyle\int_0^\infty\frac{x^{k-1}}{(x^n+a^n)^m}dx~$ can be evaluated by substituting $x=at$ and $u=\dfrac1{t^n+1}$ , then recognizing the expression of the beta function in the new integral, and lastly 
employing Euler's reflection formula for the $\Gamma$ function to simplify the result.