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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?:)

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    $\begingroup$ Use that $x+\cos(x) \geq 1$ for $x\geq 0$ togeather with the fact that $\int_0^1\frac{dx}{\sqrt{x}} = 2$ is finite. $\endgroup$ – Winther Jan 26 '16 at 22:18
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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 $$ \int_0^1\frac{dx}{\sqrt{x}(x+\cos(x))}. $$

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