Question: Discuss the convergence of the improper integral $\int\_{2}^{\infty}\frac{\sin x}{x(\ln x)^2}dx$

Answer: 
Note that we have

$$\begin{align}
\left|\int\_2^\infty \frac{\sin(x)}{x\log^2(x)}\,dx\right|&\le \int\_2^\infty \frac{|\sin(x)|}{x\log^2(x)}\,dx\\\\
&\le \int\_2^\infty \frac{1}{x\log^2(x)}\,dx\\\\
&=\frac{1}{\log(2)}
\end{align}$$
The integral is convergent by (the integral version of) [Dirichlet's test](https://en.wikipedia.org/wiki/Dirichlet%27s_test): $\sin(x)$ is a function with a bounded primitive and $\frac{1}{x\log^2(x)}$ is a positive function on $(2,+\infty)$, decreasing to zero.

Additionally,
$$ \int\_{e}^{e^M}\frac{\sin x}{x\log^2 x}\,dx = \int\_{1}^{M}\frac{\sin(e^u)}{u^2}\,du \leq \int\_{1}^{M}\frac{du}{u^2} = 1-\frac{1}{M}. $$