I know this is a silly question, but I've tried to find an answer using my TI-89 calculator, Maple and wolframalpha but none of those could tell me whether
$$\int_0^\infty |\cos(x^2)| \mathrm dx$$
converges or diverges.
Thus, I'd be very happy if someone could help me out and tell me, whether the given integral converges or not (and why?). Thanks a lot.
Best Answer
It diverges. You should be able to prove that $|\cos(x^2)|>0.1$ for most x. If you let x=$\sqrt{\pi}u$ it is easier to assess the range of u where the cosine is close to zero.