How to prove the non existence of this integral?

Question

How to prove that \begin{equation}\nonumber \int_0^\infty \sin^2\left[\pi\left(x + \frac{1}{x}\right)\right]dx \end{equation} does not exist?

Answer 1

Write it as $\int_0^\infty \frac12 (1- \cos(\pi(2x+2/x)))dx$. Even the constant factor $\frac12$ results in an infinite integral...