I am a student taking multivariable calculus and we just had an exam and I cannot get over one question in particular: $$\frac{3}{4\pi}\int\int\int\limits_\mathbb{R^3}e^{-\sqrt{x^2+y^2+z^2}^3}dV$$ I had some ideas in mind, such as maybe switching to cylindrical or spherical coordinates, but whatever I did, I still ended up with a difficult integration. I also don't think I could've used any other transformation besides cylindrical or spherical coordinate because we had gone over that one lecture before the exam and I'm pretty sure he said we wouldn't be tested on that. If that's what it takes then so be it, I just want to know how it's done.
Another piece of information that may be useful: On the exam I assumed the answer to be $1$ for two reasons; because of the coefficient, and because of a part (b) to this question asking what the integral may represent. After checking the result on WolframAlpha, I found that the answer is indeed $1$, but I still cannot find an explanation as it wouldn't let me show the steps used to solve it.
I'd really love to see the solution to this one.
using spherical coordinates you get: $$ \int_0^{2\pi}\int_0^{\pi}\int_0^{\infty} e^{-r^3}r^2\sin(\theta)dr d\theta d\phi=2\pi\int_0^{\pi}[-\frac{1}{3}e^{-r^3}]_0^{\infty}\sin(\theta)d\theta=\frac{2\pi}{3}\cdot [-\cos(\theta)]_0^{\pi}=\frac{4\pi}{3} $$
Best regards, Serdar