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'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 ..."
$\displaystyle I_{n+1} = \int \frac{\cos^{2(n+1)} x}{\sin x} ~dx = \cos^2 x \int \frac{\cos^{2n} x}{\sin x} ~dx = I_n \cos^2 x = I_n - I_n \sin^2 x$
$\displaystyle I_n \sin^2 x = \int \frac{\sin^2 x\cos^{2n} x}{\sin x} ~dx = \int \sin x \cos^{2n} x ~dx = - \frac{\cos^{2n+1} x}{2n + 1}$
hence,
$\displaystyle I_{n+1} = I_n - I_n \sin^2 x = I_n + \frac{\cos^{2n+1} x}{2n + 1}$
and the result follow