Convergence of the integral $\int \limits ^\infty _ 0 \frac{dx}{\sqrt{x}(x+\cos(x))}$

Question

My question is how to prove the convergence of integral $$\int \limits ^\infty _ 0 \frac{dx}{\sqrt{x}(x+\cos(x))}$$ I already have, that $\int \limits ^\infty _ 1 \frac{dx}{\sqrt{x}(x+\cos(x))}$ converges by using the substitution $y=\sqrt{x}$ and the majoring sum $\sum \limits ^{\infty}_{n=2}\frac{1}{n^2-1}$, but I have no idea for the first part of the integral. Any suggestions?:)

Answer 1

One may observe that, as $x \to 0^+$, $$ \frac1{\sqrt{x}(x+\cos(x))}\sim \frac1{\sqrt{x}(x+1)} \sim \frac1{\sqrt{x}} $$ and the latter integrand is integrable in a neighborhood of $0$, giving the convergence of $$ \int_0^1\frac{dx}{\sqrt{x}(x+\cos(x))}. $$