What is a cookbook or cheat book for this types of multi-variable integrals? I saw somewhere the following integral:
$$ \int_{\mathbb{R}^d} \Vert x\Vert^3 e^{-a\Vert x\Vert^2}dx=a^{-\frac{M+3}{2}}\pi^{\frac{m}{2}}\frac{\Gamma(\frac{m+3}{2})}{\Gamma(\frac{m}{2})}$$
What is a source which lists this types of integrals? A kind of cookbook, which we can look up for this types of multivariable integrals?
If there isn't any handbook, how to solve it?
I searched the Internet, but didn't find any thing. 
 A: In polar coordinates, we have the following formula for the volume form:
$dV = r^{n-1} sin^{n-2} ( \phi_1) \ldots sin(\phi_{n-2}) dr d\phi_1 \ldots d\phi_{n-1} = r^{n-1} S dr d\Phi$, where $d\Phi = d\phi_1 \ldots d\phi_{n-1}$ and $S$ is the function of $\phi_1, \ldots, \phi_{n-2}$ appearing in the integrand.
See: https://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates
Let $f(x) = ||x||^3 exp( - a||x||^2) = r^3 exp( - ar^2)$ 
Hence $\int_{R^{n}} f(x) dx =  \int_{S^{n-1}} ( \int_0^{\infty} f(r) r^{n-1} dr ) S d \Phi = ( \int_0^{\infty} f(r) r^{n-1} dr ) (\int_{S^{n-1}} S d\phi) = ( \int_0^{\infty} f(r) r^{n-1} dr ) (\text{surface area of n-1 sphere})$.
Wolfram tells us $\int_0^{\infty} f(r) r^{n-1} dr = \int_0^{\infty} r^{n + 2} exp(- ar^2) dr = \frac{1}{2} a^{-n/2 - 3/2} \Gamma((n + 3)/2)$: http://www.wolframalpha.com/input/?i=integrate+r%5E3+e%5E%7B-ar%5E2%7D+r%5E%7Bn-1%7D+from+0+to+infinity
(You could probably do this integral by hand.)
Wolfram also knows the surface area of the sphere: http://www.wolframalpha.com/input/?i=surface+area+of+n+dimensional+sphere
Multiplying the two gives us the formula in your question: here
Let me know if any steps are unclear. :)
