Integral: Solving sin(101x) sin^99(x) dx

[icystrike]

Homework Statement

\int sin(101x) sin^99(x) dx

Homework Equations

Complex Number

The Attempt at a Solution

sin(101x) = \frac{e^{101ix}-e^{-101ix}}{2i}
sin^99(x) = Im(e^{99ix})

Still trying...

 

[Mr.Miyagi]

\sin x=\frac{e^{ix}-e^{-ix}}{2i}
Does that help?

edit: Ah, sorry. Didn't see the mangled tex. just a minute.

 

[Mr.Miyagi]

There is an identity for sin^nx which transforms it into a sum of regular sines. Perhaps that is a place to start.

 

[lurflurf]

use reduction formulae
try an identity from elementary trigonometry such as
sin(101x)sin(9x)^9=[exp(101 i x)-exp(-101 i x)][exp(9 i x)-exp(-9 i x)]^9/2^10
from which (or otherwise) one may see that
sin(101x)sin(9x)^9=(1/512)(cos(20 x)-9 cos(38 x)+36 cos(56 x)-84 cos(74 x)+126 cos(92 x)-126 cos(110 x)+84 cos(128 x)-36 cos(146 x)+9 cos(164 x)-cos(182 x))