Convergence of $\int_0^1\frac{1}{t^{\beta}}\sin(\frac{\pi}{t})dt$

Question

Question: Check if the integral converges $$\int_0^1\frac{1}{t^{\beta}}\sin\left(\frac{\pi}{t}\right)dt$$ I have checked that if $\beta<1$ then the integral is absolutely convergent and if $\beta=1$ it is convergent (but not absolutely). I got stuck at the case $\beta>1$. Any help is appreciated.

Answer 1

Hint: You can reduce this to a more familiar integral by setting $x = \pi/t$.