
need help with reduction formula

Quote:
Originally Posted by
justin016 for the last part, if I tranpose the last integral to the left side, the constant, should it still be (n1)? why does it changes to n instead?
Moving the rightmost integral in your penultimate line to the lefthand side, you have
$\displaystyle \int \sin^nx\,dx+(n1)\int\sin^nx\,dx=\sin^{n1}x\cos x+(n1)\int\sin^{n2}x\,dx$
$\displaystyle \implies(1+n1)\int\sin^n x\,dx = \sin^{n1}x\cos x + (n1)\int\sin^{n2}x\,dx$
$\displaystyle \implies n\int\sin^nx\,dx = \sin^{n1}x\cos x + (n1)\int\sin^{n2}x\,dx$
$\displaystyle \implies \int\sin^nx\,dx = \frac{\sin^{n1}x\cos x}{n} + \frac{n1}{n}\int\sin^{n2}x\,dx.$