# Testing for convergence. (Improper Integral)

How can I test this integral or convergence:

$$\int_1^\infty \frac{2x-1}{\sqrt{x^5 + 2x - 2}} dx$$

I'm trying to find integral of higher function and in result i get divergence, so I cant use this information.

## 1 Answer

First, $x^5+2x-2>0$ for $x\geq 1$ so we do not need to worry about infinite discontinuity of the integrand within the interval of integration. Next, noting that the numerator is of degree 1 and the denominator of degree $\frac{5}{2}$, we compare the integrand with $x^{-3/2}$ and apply the limit comparison test.

• x^(-3/2) is convergent and less than integrand, so comparison test doesn't work for me. – Rasul Kerimov May 11 '15 at 12:36
• I said we should use limit comparison test instead of direct comparison test which you implicitly referred to. There is no need to worry about which one is larger when using limit comparison because it only cares about the declining rate of the two functions. – Alex Fok May 11 '15 at 12:40