For what $a$ is this integral convergent?$\int_{0}^{\frac{\pi}{2}} \log(\cos(x))*(\tan(x))^a dx$, $a \in \mathbb{R}$

Question

I’m new here and I’m trying to solve this exercise:

For what $a$ does the integral converge?

$$\int_{0}^{\frac{\pi}{2}} \log(\cos(x))*(\tan(x))^a dx$$ $$a \in \mathbb{R}$$

So far I got to

$$\int_{0}^{1} \frac{\log(t)}{t^a} dt$$

(by Limit Comparison Test with

$$\int_{0}^{\frac{\pi}{2}} \log(\cos(x))*\frac{\sin(x)}{(\cos(x))^a} dx$$ and subbing $t= \cos(x)$) but I don't know how to solve this. I would appreciate any help!

Answer 1

For $a\geq 1$, we have \begin{align*} \int_{0}^{1}-\dfrac{\log t}{t^{a}}dt&=\int_{0}^{1}\dfrac{\log(1/t)}{t^{a}}dt\\ &\geq\int_{0}^{1/3}\dfrac{\log(1/t)}{t^{a}}dt\\ &\geq\int_{0}^{1/3}\dfrac{1}{t^{a}}\\ &=\infty. \end{align*}