Given the improper integral below,
$$ I = \int_{-\infty}^{+\infty} x^4e^{-x^4}dx $$
Is it correct that this is an even function since $x^4 \cdot e^{-x^4}$ is just a product of two even functions? If so, does this mean that I can also be equal to
$$ I = 2 \int_{0}^{+\infty} x^4e^{-x^4}dx$$
This is what I know so far and I'm still having a hard time on what to do next. Can anyone teach me on one of the approaches that I can make to evaluate this integral?
You are right about even function property. If you substitute $t=x^4$, you will have an integral that suspiciously resembles the Gamma function.