TITLE: Does $\int_{0}^{\infty} \sin^x(x) dx$ converge?
QUESTION [10 upvotes]: From what I have found the indefinite integral does not have a closed form solution. Also, the function takes complex values except for in the intervals $[0,\pi],[2\pi,3\pi],$ (and for whole x between those intervals). But if we only considered those intervals where the function takes on real values, i.e.
$$\int_{0}^{\pi}\sin^x(x)dx+\int_{2\pi}^{3\pi}\sin^x(x)dx+\int_{4\pi}^{5\pi}\sin^x(x)dx+{...}$$
Does this infinite sum converge? For the record I have no idea how to go about proving this, but I am curious if anyone does.

REPLY [2 votes]: If your integral converges also
$$\sum_{n=0}^\infty \int_0^1 x^n dx = \sum_{n=1}^\infty \frac{1}{n}= \infty$$
would converge. This is because for $x \in [0,\pi/2]$ we have
$$\frac{2x}{\pi} \le \sin(x).$$