Is the integral $\int^{\infty}_{0}\frac{\arctan x}{x}dx$ convergent?

Question

I was given this integral:

$$\int^{\infty}_{0}\frac{\arctan(x)}{x}dx$$

As the title says, I have to find out whether it is convergent or not. So far, I have tried integrating by parts and substituting ${\arctan(x)}$, and neither got me anywhere.

Answer 1

If $\operatorname{atg}$ means the inverse tangent function, then your integral does not converge.

One can see that $\arctan x\ge \pi/4$ if $x\ge 1$, so $$\frac{\arctan x}{x}\ge \frac{\pi}{4x}$$ for $x\ge 1$. Since $\int_1^\infty\frac{\pi}{4x}dx$ diverges, so does $\int_1^\infty\frac{\arctan x}{x}dx$.

Answer 2

$$\int_0^\infty\frac{\arctan(x)}{x}dx=\int_{-\infty}^\infty\arctan(e^t)\,dt$$ but this integrand does not vanish.