Is the integral test for convergence still applicable?

$$\sum _{n=0}^{\infty \:}\left(n\ e^{-n^2}\right)$$

Can I still use the integral test to determine whether this series converges or diverges given that $f(x) = x\ e^{-x^2}$ is not decreasing on the interval $(0,\frac{\sqrt{2}}{2}]$?