Prove the following:

Question

1. sin³xcos³x = (cos³x - cos⁵x)sinx

2. Hence or otherwise determine:

∫ sin³xcos³x dx

Answer 1

Use u-substitution.

Let u=sin x,

then du=cos x dx.

∫sin³xcos³x dx

= ∫sin³x cos²x cos x dx

= ∫sin³x (1-sin²x) cos x dx

= ∫u³(1-u²) du

= ∫(u³-u⁵) du

= 1/4u⁴-1/6u⁶+C

= 1/4sin⁴x-1/6sin⁶x+C