# Finding whether the following integral converges or diverges?

May 19, 2018

The integral converges

#### Explanation:

This is the graph of the function $y = \frac{1}{\sqrt{{x}^{5} + 2}}$.

Clearly, from $x = 1$ onwards, the area under the curve converges to a finite value.

May 19, 2018

Obviously

${\int}_{1}^{\infty} \frac{1}{\sqrt{{x}^{5} + 2}} \mathrm{dx} > 0$

For $x > 1$ we have

${x}^{4} < {x}^{5} + 2 \implies \frac{1}{x} ^ 2 > \frac{1}{\sqrt{{x}^{5} + 2}}$

Thus

${\int}_{1}^{\infty} \frac{1}{\sqrt{{x}^{5} + 2}} \mathrm{dx} < {\int}_{1}^{\infty} \frac{\mathrm{dx}}{x} ^ 2$

The latter integral is

${\lim}_{L \to \infty} {\int}_{1}^{L} \frac{\mathrm{dx}}{x} ^ 2 = {\lim}_{L \to \infty} \left(1 - \frac{1}{L}\right) = 1$

and thus we have

$0 < {\int}_{1}^{\infty} \frac{1}{\sqrt{{x}^{5} + 2}} \mathrm{dx} < 1$

and thus the integral converges.