Integral of sin x sin 2x?

Ok, I know that sin 2x can be substituted out for 2 sin x cos x.

so now I have the Integral of (sin x ) ( 2 sin x cos x ) dx which is 2 sin2x cos x dx

If I use integration by parts with the u and dv, I find myself right back again..can you help. The book gives an answer, but I am not sure how it was achieved.

the book gives 2/3 sin 2x cos x - 1/3 cos 2x sin x + C I may be making this more difficult than it is ?

when you get the integral of 2 sin2x cos x dx use u substitution. u= sinx du= cosxdx. Then you'll get the integral of 2u^2 du.. .and then integrate...