Find all $\alpha\in\mathbb{R}$ such that integral converges.

Question

Find all parameters $\alpha\in\mathbb{R}$ such that $$ \int_{1}^{+\infty}x\cdot\cos^{\alpha}\left(\frac{\pi}{2}\cdot\frac{x+1}{x+2}\right)\space dx$$ converges.

Answer 1

You have $$x \cos^{\alpha} \left( \frac{\pi}{2}. \frac{x+1}{x+2}\right) = x\cos^{\alpha} \left( \frac{\pi}{2}. \left(1-\frac{1}{x+2}\right)\right)=x\sin^{\alpha} \left( \frac{\pi}{2x+4}\right) $$

When $x$ tends to $+\infty$, you have $$x\sin^{\alpha} \left( \frac{\pi}{2x+4}\right) \sim x \left( \frac{\pi}{2x+4}\right)^{\alpha} \sim \left(\frac{\pi}{2}\right)^{\alpha}x^{1-\alpha} $$

You deduce that the integral converges if and only if $x^{1-\alpha}$ is integrable in $+\infty$, i.e. if and only if $\alpha > 2$.