Determine if the integral is divergent or convergent

Question

I should determine whether this is a convergent or divergent integral. The problem is that I don't know how to start. $$ \int_{1}^{+\infty} \frac{x\ \sin(x)\ dx}{\sqrt{1+x^5}} $$

Answer 1

Note that $$\left \vert\dfrac{x \sin(x)}{\sqrt{1+x^5}} \right \vert \leq \dfrac{x}{\sqrt{1+x^5}} \leq \dfrac{x}{x^{5/2}} = \dfrac1{x^{3/2}}$$ Now you should be able to finish it off.