TITLE: Simpler way to compute a definite integral without resorting to partial fractions?
QUESTION [38 upvotes]: I found the method of partial fractions very laborious to solve this definite integral :
$$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$
Is there a simpler way to do this ?

REPLY [2 votes]: Another way
Let 
$$ I= \int_0^\infty \frac {x^{1/3}\, \mathrm{d}x}{x^2 + 1}. $$
By the Schwinger parametrization we have
$$ I=\int_0^\infty \mathrm{d}t\, \exp(-t)\int_0^\infty \mathrm{d}x\, x^{1/3}\, \exp\left(-tx^2\right).$$
The last integral can be calculated along the lines of this answer. Using this result, one gets
$$I=\int_0^\infty \mathrm{d}t\,\frac{ \Gamma \left(\frac{2}{3}\right)\exp(-t)}{2 t^{2/3}}=\frac{1}{2}\Gamma \left(\frac{2}{3}\right)\Gamma \left(\frac{1}{3}\right)=\frac{\pi}{\sqrt{3}},$$
since
$$ \Gamma\left(\nu,x\right)\equiv\int_x^\infty \mathrm{d}t\, t^{\nu-1}\exp(-t).$$