Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

Question

I've tried to partial integrate this integral from $0$ to $1$, $1$ to $e$, and $e$ to infinity:

$$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx.$$

Without success. Any help?

Answer 1

The integral is convergent by Dirichlet's test, since $\cos x$ has a bounded primitive while $\frac{1}{\sqrt[3]{\ln x}}$ is decreasing towards zero. Integration by parts gives: $$ \int_{e}^{M}\frac{\cos x}{\sqrt[3]{\log x}}\,dx = \left.\frac{\sin x}{\sqrt[3]{\ln x}}\right|_{e}^{M}+\int_{e}^{M}\frac{\sin x}{3x\ln x\sqrt[3]{\log x}}\,dx$$and $$ \int_{e}^{+\infty}\frac{dx}{3x\,\left(\ln x\right)^{4/3}} = 1.$$