Investigate the convergence of $$\int_0^\infty \frac{\sin x^2}{x} \, \mathrm{d}x$$ Is it converging? Converging absolutely?
I want to use Dirichlet's test for integrals. Let $f(x) = \frac 1 x$ and $g(x) = \sin x^2$. $f(x)$ is decreasing to $0$. Now, I need to show that there's an $M > 0$ such that $\int_0^\omega \sin x^2 \, \mathrm{d}x \le M$ for every $\omega$.
How do I do that? And what should be done for the "converging absolutely" part?
Thanks.
Besides @MichaelBurr 's very smart idea exposed in a comment under the original post, you could also use the change of variable $t = x^2$ which transforms your integral into $\frac 1 2 \int \limits _0 ^\infty \frac {\sin t} t \Bbb d t$, to which you can now immediately apply the Abel-Dirichlet theorem.