does $\int _0^{\infty }\frac{\sin\pi \:x}{\left|\ln \left(x\right)\right|^{\frac{3}{2}}}$ converge?

Question

that's a question from some exam in Calculus Can someone help?

does $\int _0^{\infty }\frac{\sin\pi \:x}{\left|\ln \left(x\right)\right|^{\frac{3}{2}}}$ converge?

I proved that it converges between 1 and infinity using comparison test with the integral of $\frac{1}{x^{\frac{3}{2}}}$ Between 1/2 and 1 i used Dirichlet exam to prove it converges. Is that true?

Any thoughts aout how can I prove between 0 and 1/2?

Answer 1

Hint:

For $0 < x < 1$ use $|\ln x| > 1-x$ and $\sin x /x < 1$ to estimate

$$\frac{\sin \pi x}{|\ln x|^{3/2}} = \frac{\pi(1-x)}{|\ln x|^{3/2}}\frac{\sin \pi x}{\pi(1-x)} = \frac{\pi(1-x)}{|\ln x|^{3/2}}\frac{\sin \pi (1-x)}{\pi(1-x)}\leqslant \frac{\pi}{\sqrt{1-x}} $$