The question:
$\displaystyle \int_{-\infty}^{\infty} x^3e^{-x^4} dx$
I tried integration by parts, but it quickly became a long mess. :/
How should one attempt this question? Thanks!
Let $\displaystyle u = -x^4$, then: $\displaystyle dx = -\frac{du}{4x^3}$
$\displaystyle \displaystyle \Rightarrow \int x^{3}{e^{-4x}\;{dx} = -\frac{1}{4}\int\frac{x^3e^u}{x^3}\;{du} = -\frac{1}{4}\int{e^u}\;{du} = -\frac{1}{4}e^u+k = -\frac{1}{4}e^{-4x}+k$
That's the antiderivative -- now do your thing with the limits.