Hi Ascher,

Write sin³(2x) as sin²(2x) sin(2x) and then use the fact that sin²(y) + cos²(y) = 1.

Write back if you need more help,

Penny

Remember how recipes say "set one egg aside"? Well, set one of the sines aside:

integral of sin² (2x) sin(2x) dx

Now use the Pythagorean identity to rewrite in cosines:

integral of (1 - cos²(2x)) sin(2x) dx

and integrate by substitution, using u = cos(2x), (-1/2)du = sin(x)dx. This or a similar trick can be used whenever you have:

-an odd power of sine and any power of cosine
-an odd power of cosine and any power of sine
-an even power of secant and any power of tangent
-an odd power of tangent and any power of secant

Only the cases with sin^even cos^even and tan^even sec^odd need special methods.

Good Hunting!
RD