1. ## Proove

I have been working on this for a long time but with no results Could someone please solve this for me.
Thanks
KC

2. Originally Posted by igodspeed

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 $x_1=r \cos \phi_1, \ x_j=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{j-1} \cos \phi_j, \ \ 2 \leq j \leq n-2,$ and

$x_{n-1}=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \cos \theta, \ x_n=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \sin \theta,$ where $0 \leq \phi_j \leq \pi, \ 0 \leq \theta \leq 2 \pi,$ and $0 \leq r \leq R.$

now find the Jacobian of this transformation and finally using gamma function your multiple integral would be pretty easy to evaluate.

3. see we did not learn the method that you are referring to, is there another way to solve it?

4. Originally Posted by igodspeed

see we did not learn the method that you are referring to, is there another way to solve it?
there's also an inductive solution. see here.

5. 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 $\mathbb{R}^n$, integration by parts, reduction formula for integrals, and being brave

6. Originally Posted by igodspeed
I have been working on this for a long time but with no results Could someone please solve this for me.
Thanks
KC
Thread closed due to OP's habit of deleting posts after getting replies.