How to prove the convergence of an integral? I need to prove the convergence of the integral $$\int_{1}^{\infty} \frac{x}{(5x^{2}-1)^3}dx $$
I tried all the tests but could not prove its convergence. I was trying the comparison test and was able to prove that $\frac{x}{(5x^{2}-1)^3} \leq \frac{x}{5x^{2}-1}$ but that wasn't much of help. Any suggestions on how to prove this integral to be convergent?
Do a standard substitution: $u = 5x^2-1 \implies xdx = \dfrac{du}{10}\implies I = \displaystyle \int_{4}^\infty \dfrac{du}{10u^3}=...$ I hope you can find the answer from this point.
If you substitute $v=5x^2-1$ you get an easier integrand (fro some constant $a$ after substitution):
$$
a\int_{1}^{\infty}\frac{dv}{v^3}
$$
which converges