Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

Question

I tried to prove convergence this way:

Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$.

$\lim_{x\to 0}\frac{f(x)}{g(x)}=1$,therefore $\int_0^1 f(x)dx$ converges iif $\int_0^1g(x)dx$ coverges,$\left|\frac{\cos\frac1x}{\sqrt x}\right|\le \frac1{\sqrt x}$,$\int_0^1\frac1{\sqrt x}dx$ converges, therefore $\int_0^1g(x)dx$ converges.

Am I right ?

Thanks.

Answer 1

This looks correct. Another way you could do it is control $|\sin{x}|$ by $|x|$ and $\cos(1/x)$ by $1$, which gives you the same integral comparison.