Need help solving - $ \int (\sin 101x) \cdot\sin^{99}x\,dx $ I have a complicated integral to solve.
I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts.
are there any simpler methods to evaluate this integral?
$$
\int\!\sin (101x)\cdot\sin^{99}(x)\, dx
$$

Note that: $$\sin(101x)=\sin(x)\cos(100x)+\cos(x)\sin(100x)$$ $$\Longrightarrow \sin(101x)\sin^{99}(x)=\sin^{100}(x)\cos(100x)+\cos(x)\sin(100x)\sin^{99}(x)$$ $$=\frac{1}{100}\left(100\cos(100x)\sin^{100}(x)+\sin(100x)(100\sin^{99}(x)\cos(x))\right)$$Is this the derivative of something?
