Integrating a jacobian to find the volume. I want to solve the following:
Prove that
$$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2} = \frac{2\pi^{n/2}}{\Gamma(n/2)}$$
where $ R=[0,2\pi] \times [0,\pi]^{n-2}$. Hint: Compute $ \int_{\mathbb R^n}e^{-|x|^2}dx$ in spherical coordinates.
So I am having problems to calculate the latter integral in spherical coordinates because I dont know how to integrate (in finite steps) $sin^{n}(x)$ and I dont know how this results to be a division of integrals.Can you help me to solve this please?, Thanks a lot in advance :)

Hint all the variables $\phi_i$ can be separted in the integral, so you can transform the integral into:
$$\displaystyle \int_R \sin^{n-2}\phi_1 \sin^{n-3}\phi_2\cdots\sin \phi_{n-2} d\theta d\phi_1\cdots d\phi_{n-2}=\int_{0}^{2\pi}d\theta \prod_{i=1}^{n-2}\int_{0}^{\pi}\sin^i\phi_i d\phi_i$$
