# Determine whether $\int _{1}^{\infty}\frac{x\sin\left(x\right)}{\sqrt{1+x^5}}\,{\rm d}x$ is convergent or divergent

My task is to determine if the following function is convergent or divergent.

$$\int_{1}^{\infty}{x\sin\left(x\right) \over\sqrt{1 + x^{5}\,}\,}\,{\rm d}x$$

I have been trying to find a more easily integrable function which could be determined to belong in any of the two categories, but I have yet to succeed. Any help is appreciated!

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Hint: For all $x\in [1+\infty[$, $$\left\vert \dfrac{x\sin{x}}{\sqrt{1+x^5}}\right\vert\leq \dfrac{1}{x^{3/2}}.$$ –  Git Gud Nov 10 '13 at 23:32
@GitGud Helped a lot, thanks! –  Jimmy C Nov 11 '13 at 0:31

## 1 Answer

Hint:$$\left\vert\frac{x\sin{x}}{\sqrt{1+x^5}}\right\vert \leq\frac{x}{\sqrt{1+x^5}}$$

What does this bound tell us about the convergence?

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$\frac{x}{\sqrt{1+x^5}}\sim \frac{x}{x^{5/2}}$. –  Mhenni Benghorbal Nov 10 '13 at 23:32