I´m trying to solve this problem about integral convergence, and I would be happy for any help.
I shoul find out for what values of $a$ is this integral convergent:
$$\int_0^\infty \text{arccot}^ax\,\cos{x}\,dx $$
Near the zero, there is no problem.
The problem is near the infinity - as the $ lim_{x\to\infty} \ arccotg^a{x} \ = 0$ if and only if $a>0$, this function is also monotonic on the inteval $[0,\infty)$ and the primitive function of $cos{\ x}$ is bounded, according to Dirichlet criterion, the integral is convergent for any $a>0$.
However, I really don't know how to show that this integral is divergent for $a \le 0$.
I've spent quite a lot of time on it, however, I have no idea what should I do.
As I've said, I would be really happy for any help or hint.
Sorry for my bad english and thanks a lot!
For large $x$, $\operatorname{arccot}{x}$ is asymptotic to $1/x$ (because $\cot{y} \sim 1/y$ for $y \to 0$). Hence, for $x \to \infty$, the integral converges only if $$ \int_{N}^{\infty} \frac{\cos{x}}{x^a} \, dx $$ does. For $a \leqslant 0$, the integrand here oscillates, with oscillations that increase in magnitude. Hence the integral does not have a finite value (write it as an alternating series and note that the terms do not converge to zero, so the series diverges).