How do I evaluate the indefinite integral sin3(x)cos2(x)dx ?

1 Answer
Sep 21, 2014

The answer is cos3x3+cos5x5+C.

The trick with sinusoidal powers is to use identities so that you can have sinx or cosx with a power of 1 and use substitution.

In this case, it is easier to get sinx to a power of 1 using sin2x=1cos2x.

sin3xcos2xdx
=sinx(1cos2x)cos2xdx
=sinx(cos2xcos4x)dx
=sinxcos2xdxsinxcos4xdx

Now it is a matter of using substitution:

u=cosx
du=sinxdx

sinxcos2xdxsinxcos4xdx
=u2du+u4du
=u33+u55+C
=cos3x3+cos5x5+C