TITLE: Real VS Complex for integrals: $\int_0^\infty \frac{dx}{1 + x^3}$
QUESTION [7 upvotes]: The integral
$$\int_0^\infty \frac{dx}{1 + x^4} = \frac{\pi}{2\sqrt2}$$
can be evaluated both by a complex method (residues) and by a
real method (partial fraction decomposition).
The complex method works also for the integral 
$$\int_0^\infty \frac{dx}{1 + x^3} = \frac{2\pi}{3\sqrt3}$$
but partial fraction decomposition does not give convergent integrals. I would like to know if there is some real method for evaluating this last integral.

REPLY [2 votes]: I would like to know if there is some real method for evaluating this last integral.

Actually, all integrals of the form $\displaystyle\int_0^\infty\frac{x^n}{1+x^m}dx$ can be solved by substituting $t=\dfrac1{1+x^m}$ , and then recognizing the expression of the beta function in the new integral, which can be written as a product of gamma functions. Then we use the reflection formula in order to finally arrive at the desired result, $I=\dfrac\pi m\cdot\csc\left[(n+1)\dfrac\pi m\right]$ — See my answer here for more information.