# A Simple Integral of a Peculiar Function

Here is problem from Mathematics Magazine (Volume 91, 2018 - Issue 1) posed by D. M. Batinetu-Giurgiu and Neculai Stanciu. Solution is by Ulrich Abel.

There is a way to find this integral actually without computing integrals at all. Well, almost.

Let $f$ be a continuous real-valued function on $(0,\infty)$ satisfying the identity $\displaystyle f\left(\frac{1}{x}\right)=-f(x)$ for all $x\gt 0.$ Given $a\gt 0,$ calculate the integral
$\displaystyle I(a)=\int_{\sqrt{2}-1}^{\sqrt{2}+1}\frac{dx}{(1+x^2)(1+a^{f(x)})}.$