I have been working on this for a long time but with no results Could someone please solve this for me.
Thanks
KC
you need to change to the generalized spherical coordinates, i.e. put $\displaystyle x_1=r \cos \phi_1, \ x_j=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{j-1} \cos \phi_j, \ \ 2 \leq j \leq n-2,$ and
$\displaystyle x_{n-1}=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \cos \theta, \ x_n=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \sin \theta,$ where $\displaystyle 0 \leq \phi_j \leq \pi, \ 0 \leq \theta \leq 2 \pi,$ and $\displaystyle 0 \leq r \leq R.$
now find the Jacobian of this transformation and finally using gamma function your multiple integral would be pretty easy to evaluate.
Wow !
I finished this pdf in around 2 hours... That was long (hey I'm still a beginner at creating pdf lol). Tell me if there is any mistake.
I hope this will help ~
Prerequisites : Integral and transformation in $\displaystyle \mathbb{R}^n$, integration by parts, reduction formula for integrals, and being brave