improper integral of exponential function

Question

I have a problem calculating improper integrals, this one for example, can you please help me solve it? $$\int_0^\infty t^3(e^{-t^2})dt$$ thanks in advance.

Answer 1

hint: Let $u = t^2, t^3dt = t^2\cdot tdt$. Can you continue?

Answer 2

Hint: Write $u = t^2$, so that ${\rm d}u = 2t\,{\rm d}t$. Rewrite $t^3e^{-t^2}$ as $-\frac{1}{2}(-2t)t^2e^{-t^2}$ , apply the substituition and integrate by parts.

Answer 3

By setting $t=\sqrt{u}$ we have $dt = \frac{1}{2}u^{-1/2}\,du$ and: $$\int_{0}^{+\infty}t^3 e^{-t^2}\,dt = \frac{1}{2}\int_{0}^{+\infty} u e^{-u}\,du = \frac{1}{2}\left.(u+1)e^{-u}\right|_{0}^{+\infty}=\color{red}{\frac{1}{2}}.$$