<s>[INST] MIT Integration Bee $21$ Question:
$$\int_0^\infty {{1}\over{(1+x^{2015})({1+x^2}})}dx$$
How to solve these integrals. I have no idea.
 [/INST] Use the substitution $x\mapsto 1/x$:
\begin{align*}
I &= \int_0^\infty {{1}\over{(1+x^{2015})({1+x^2}})}dx \\
&= \int_0^\infty {{1}\over{(1+x^{-2015})({1+x^{-2}}})} \frac{1}{x^2}dx\\
&= \int_0^\infty {{x^{2015}}\over{(1+x^{2015})({1+x^{2}}})} dx
\end{align*}
Then, 
\begin{align*}
2I &= \int_0^\infty {{1}\over{(1+x^{2015})({1+x^2}})}dx+\int_0^\infty {{x^{2015}}\over{(1+x^{2015})({1+x^2}})}dx\\
&=\int_0^\infty {{1+x^{2015}}\over{(1+x^{2015})({1+x^2}})}dx\\
&=\int_0^\infty \frac{1}{1+x^2}dx = \frac{\pi}{2}
\end{align*}
So $I=\pi/4$.
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