For what values of $a$ does $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{x})^a \ dx$ converge?

Question

For what values of $a$ does $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{x})^a \ dx$ converge? $a\in \mathbb{R}$

How to approach this problem?

EDIT: For $a\ge 0$ the $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{x})^a \ dx$ converges. The only problem is for $a<0$. My guess for $-1<a<0$ the $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos{x})^a \ dx$ converges and for $a\le-1$ it doesn't. I'm not sure how to approach this formally.

Thanks in advance.

Answer 1

Your first consideration is fine, we need to consider the case $a<0$, then since we have

• at $x=\frac \pi 2$

$$\cos x = \left(\frac \pi 2 - x\right)+O(x^3)$$

• at $x=-\frac \pi 2$

$$\cos x = \left(\frac \pi 2 + x\right)+O(x^3)$$

the naure of the given integral at these points can be compared with $\int_{0}^{\frac{\pi}{2}} \left(\frac \pi 2 - x\right)^a \ dx$ and $\int_{-\frac{\pi}{2}}^{0} \left(\frac \pi 2 + x\right)^a \ dx$, therefore also your guess looks fine and you can use this observation to formalize your conclusion.