Improper integral of $\frac{1}{x} \sin \frac{1}{x}$

Question

Let $f(x)= \frac{1}{x} \sin \frac{1}{x}$ when $x \in (0, 1]$and $f(0)=0$. Prove that $\int_{0}^1 f(x)dx$ exists.

Can someone give me a hint to solution?

Answer 1

Hint: Make the change of variable $x = 1/y.$