We have,
$P(n): \displaystyle \int \sin^{n}(x)dx = -\frac{1}{n}\sin^{n-1}(x)\cos(x)+\frac{n-1}{n}\int \sin^{n-2}(x)dx$
$P(n+1): \displaystyle \int \sin^{n+1}(x)dx = -\frac{1}{n+1}\sin^{n}(x)\cos(x)+\frac{n}{n+1}\int \sin^{n-1}(x)dx$
$\displaystyle \int \sin(x)\sin^{n}(x)dx.$
$u=\sin^{n}(x) \Rightarrow du=n\sin^{n-1}(x)\cos(x)dx$
$dv=\sin(x) \Rightarrow v=-\cos(x)$
So,
$\displaystyle \int \sin^{n+1}(x)dx= -\sin^{n}(x)\cos(x)+n\int \cos^{2}(x)\sin^{n-1}(x)dx$
Now, I'm having difficulty applying the $P(n)$ induction hypothesis. If anyone can contribute to this, I would appreciate it.