Math Help - Integration - Reduction Method: PLEASE HELP!!!

1. Integration - Reduction Method: PLEASE HELP!!!

I've attempted both questions but I cant seem to get the answers. Could someone please show me where I got it wrong.

P.S. For the question that's typed up, I'm trying to find out the last part of the question i.e. "evaluate ..."

2. Originally Posted by xwrathbringerx
I've attempted both questions but I cant seem to get the answers. Could someone please show me where I got it wrong.

P.S. For the question that's typed up, I'm trying to find out the last part of the question i.e. "evaluate ..."
$I_{n+1} = \int \frac{\cos^{2(n+1)} x}{\sin x} ~dx = \cos^2 x \int \frac{\cos^{2n} x}{\sin x} ~dx = I_n \cos^2 x = I_n - I_n \sin^2 x$

$I_n \sin^2 x = \int \frac{\sin^2 x\cos^{2n} x}{\sin x} ~dx = \int \sin x \cos^{2n} x ~dx = - \frac{\cos^{2n+1} x}{2n + 1}$

hence,

$I_{n+1} = I_n - I_n \sin^2 x = I_n + \frac{\cos^{2n+1} x}{2n + 1}$

and the result follow

3. I cant seem to be able to get the next part at all...