Tools for investigating the convergence of the improper integral $\int_0^1\cos(1/x)\mathrm dx$ and similar integrals

Question

I am struggling to see why the integral $$ \int_0^1 \cos(1/x)\mathrm dx $$

Is convergent and why for example $$ \int_0^1 \frac{1}{x}\cos(1/x)\mathrm dx $$ Is only conditionally convergent.

My usual tools like taylor expansion, investigating the limit of the integrand multiplied by $x^p$ (quotient test) don't apply. Furthermore, I am not sure how to tackle the limit since these Is there something clever I can bound these integrals below by?

I have been able to see why $$ \int_0^1 \frac{1}{x^2}\cos(1/x)\mathrm dx $$ Is divergent, since that has an elementary antiderivative.

Answer 1

Let $I$ be the integral given by

$$I=\int_0^1 \cos(1/x)\,dx \tag 1$$

Now, enforcing the substitution $x\to 1/x$ in $(1)$ yields

$$I=\int_1^\infty \frac{\cos(x)}{x^2}\,dx \tag 2$$

Since the integral in $(2)$ absolutely converges, it converges. Therefore, the integral in $(1)$ converges.

Answer 2

If $f:[0,1] \to \mathbb R$ is bounded and $f$ is Riemann integrable on $[a,1]$ for $0< a <1,$ then $f$ is Riemann integrable on $[0,1].$ This shows$\int_0^1 \cos (1/x)$ converges.