for the last part, if I tranpose the last integral to the left side, the constant, should it still be (n-1)? why does it changes to n instead?
Moving the rightmost integral in your penultimate line to the left-hand side, you have
$\displaystyle \int \sin^nx\,dx+(n-1)\int\sin^nx\,dx=-\sin^{n-1}x\cos x+(n-1)\int\sin^{n-2}x\,dx$
$\displaystyle \implies(1+n-1)\int\sin^n x\,dx = -\sin^{n-1}x\cos x + (n-1)\int\sin^{n-2}x\,dx$
$\displaystyle \implies n\int\sin^nx\,dx = -\sin^{n-1}x\cos x + (n-1)\int\sin^{n-2}x\,dx$
$\displaystyle \implies \int\sin^nx\,dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\,dx.$