About the convergence of the improper integrals $\int_0^{\pi/2} \frac{1}{e^x - \cos x} dx$, $\int_0^{\pi} \frac{1}{\cos \alpha - \cos x} dx$

Question

About the convergence of the improper integrals:

1: $\int_0^{\pi/2} \frac{1}{e^x - \cos x} dx$

2: $\int_0^{\pi} \frac{1}{\cos \alpha - \cos x} dx, 0 \leq \alpha \leq \pi.$

In the first problem $0$ is a point of infinite discontinuity.

Having problem in verifying the convergence. Help please.

Answer 1

Hoping that I am not off-topic, consider (for the first problem) Taylor expansionsaround $x=0$; it gives $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+O\left(x^4\right)$$ $$\cos(x)=1-\frac{x^2}{2}+O\left(x^4\right) $$ $$e^x-\cos(x)=x+x^2+\frac{x^3}{6}+O\left(x^4\right)$$ Perform the long division $$\frac 1 {e^x-\cos(x)}=\frac{1}{x}-1+\frac{5 x}{6}+O\left(x^2\right)$$ I am sure that you see what is going on and that you can take it from here.