I tried doing :
$\int_{1}^{\infty}\ \frac{\sqrt[3]{x-1}}{\sqrt{x^3+3}}dx \approx \int_{1}^{\infty}\ \frac{x^\frac{1}{3}}{x^\frac{3}{2}}dx = \int_{1}^{\infty}\ \frac{1}{x^\frac{7}{6}}dx $
According to the P test for improper integrals , we get :
$ \frac{7}{6} >1$
$\Longrightarrow $ The Integral $converges$ .
I'm not sure if my solution is valid and works , However according to an online computing website the integral does in fact $converge$.
I Would like to know if there is another way to solve this integral and if my math works !
Thanks in Advance !