Determining if an integral is absolutely convergent

Question

Any idea why the following integral is conditionally convergent and not absolutely convergent?

$$\int_{1}^{\infty} \frac{\sin(3x)\cdot \arctan(x)}{\sqrt{x}} dx $$

Answer 1

It is conditionally convergent by Dirichlet's test, since $\sin(x)$ has a bounded primitive on $(2,+\infty)$ while $\frac{\arctan x}{\sqrt{x}}$ is decreasing towards zero on the same interval. It is not absolutely convergent because $$\left|\sin x\right| = \frac{2}{\pi}-\frac{4}{\pi}\sum_{n\geq 1}\frac{\cos(2nx)}{4n^2-1} $$ gives that $$\int_{0}^{N}\frac{\left|\sin x\right|}{\sqrt{x}}\,dx = \frac{4}{\pi}\sqrt{N}+O(1),$$ for instance. The arctangent term does not play a major role since $\arctan(x)$ is bounded between $\frac{\pi}{4}$ and $\frac{\pi}{2}$ on $(1,+\infty)$.