The value of the definite integral $\int_{0}^{\infty}\frac{dx}{(1+x^a)(1+x^2)}$

Question

The value of the definite integral $$\int_{0}^{\infty}\frac{dx}{(1+x^a)(1+x^2)}$$ $(a>0)$ is?

MY ATTEMPTS:

1)Partial Fraction-Not possible

2)Substitutions-Could'nt think of any

3)General rules for changing limits of integration do not seem to work.

Any suggestions/hints?

Answer 1

... always the same. By the substitution $x=\frac{1}{z}$ we have

$$I(\alpha)=\int_{0}^{+\infty}\frac{dx}{(1+x^\alpha)(1+x^2)}=\int_{0}^{+\infty}\frac{dz}{(1+z^{-\alpha})(1+z^2)} $$ hence $$ 2\,I(\alpha) = \int_{0}^{+\infty}\left(\frac{1}{1+x^\alpha}+\frac{1}{1+x^{-\alpha}}\right)\frac{dx}{1+x^2}=\int_{0}^{+\infty}\frac{dx}{1+x^2}=\frac{\pi}{2}. $$ A symmetry trick suddenly appears: $I(\alpha)=\color{red}{\frac{\pi}{4}}$.