In the final question of the MIT Integration Bee $2020$ qualifier exam, the participants were asked to evaluate
$$\int_0^\infty x^5 e^{-x^4}~dx$$
and in the solutions they give the answer as $$\frac{\sqrt\pi}{8}$$
Most, if not all the other questions can be completed with the use of elementary methods and functions, but I could not solve this question. The integrand is strongly suggestive of the Gamma function and the Gaussian integral, but I am not competent with these yet, so I couldn't employ them to evaluate the integral.
So, how would I solve this integral? If you must resort to non-elementary functions and results and techniques, then so be it.
Thank you for your help.
Best Answer
$$\int_0^\infty x^5 e^{-x^4}~dx$$ Let $x^2= u$.
$$\frac12\int_0^\infty u^2 e^{-u^2}~du$$
$$\frac12\int_0^\infty u \cdot u e^{-u^2}~du$$
Now use integration by parts and use that $\int_0^\infty e^{-u^2}~du=\frac{\sqrt\pi}{2}$