Convergence of improper integral, $\cos(1/x)$

Question

I'm trying to deduce weather this improper integral is convergent or not: $$ \int_{0}^{1}\dfrac{\cos(\frac{1}{x})}{x}dx. $$ I've tried using Dirichlet's test for convergence, yet I cant seem to properly 'place' the functions under the needed terms.

I was hinted by a colleague that substitution can be applied here yet I see no way of utilizing that method.

I'm not sure how to move forward at this point, hints are happily accepted!

Thanks in advance!

Answer 1

With $t=1/x$ and by parts, you get $$\int_0^1\frac{\cos(1/x)}{x}dx=\int_1^\infty\frac{\cos t}{t}dt=\left[\frac{\sin t}{t}\right]_1^\infty+\int_1^\infty\frac{\sin t}{t^2}dt.$$