Integral $\int_{\mathbb{R}^n} \frac{1}{(2 \pi)^{\frac{n}{2}}} e^{-\frac{\| x-\mu\|^2}{2}} dx$ using Spherical coordiantes I know that, the inetegral 
\begin{align}
\int_{\mathbb{R}^n} \frac{1}{(2 \pi)^{\frac{n}{2}}} e^{-\frac{\| x-\mu\|^2}{2}} dx
\end{align}
integrates to $1$.
This can be shown as follows
\begin{align}
\int_{\mathbb{R}^n} \prod_{i=1}^n \frac{1}{\sqrt{2 \pi}} e^{-\frac{( x_i- \mu_i)^2}{2}} dx
\end{align}
and see that every inetegral is equal to $1$. 
My question is how to do this by using spherical coordinates? 
Thanks.
 A: Obviously $\mu$ is completely irrelevant: it can be removed by a suitable translation.
By Fubini's theorem
$$ \int_{\mathbb{R}^n}\exp\left(-\|x\|^2\right)\,dx = \left(\int_{-\infty}^{+\infty}e^{-x^2}\,dx\right)^n = \pi^{n/2} $$
and the claim readily follows. Have a look at the first pages of Keith Ball, An Elementary Introduction to Modern Convex Geometry. If you want to use spherical coordinates, well, just use them by computing the determinant of the involved jacobian matrix.
A: We can write the integral of interest as
$$\begin{align}
I&=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n} e^{-\frac12||\vec x-\vec \mu||^2}\,d^n\vec x\\\\
&=\frac{1}{(2\pi)^{n/2}}\int_{S_{n-1}} \color{blue}{\int_0^\infty e^{-\frac12r ^2}\,r^{n-1}\,dr}\,dS_{n-1}\\\\
&=\frac{1}{(2\pi)^{n/2}}\int_{S_{n-1}} \color{blue}{\left(\frac12 2^{n/2}\Gamma(n/2)\right)}\,dS_{n-1}\\\\
&\frac{1}{(2\pi)^{n/2}}\color{blue}{\left(\frac12 2^{n/2}\Gamma(n/2)\right)}\color{red}{\int_{S_{n-1}}(1)\,dS_{n-1}}\\\\
&\frac{1}{(2\pi)^{n/2}}\color{blue}{\left(\frac12 2^{n/2}\Gamma(n/2)\right)}\color{red}{\frac{2\pi^{n/2}}{\Gamma(n/2)}}\\\\
&=1
\end{align}$$
as was to be shown!  

A good summary for analysis of the $n$-Sphere in spherical coordinates can be found HERE.

