sin(x)^4 = [(1 - cos(2x)) / 2]^2 // half-angle identity

1/4 - cos(2x)/2 + cos(2x)^2/4 // expand

1/4 - cos(2x)/2 + 1/4((1 + cos(2x)) / 2) // half-angle identity

3/8 - cos(2x)/2 + cos(2x)/8 // simplify

3/8x - sin(2x)/4 + sin(2x)/16 // integrate

Hoping I didn't get lost along the way. =p Still don't know how to use the math notation.

Edit: Whoop, sorry, second half-angle identity should have been (1 + cos(4x)) / 2. Answer's the same, but the last term is sin(4x)/32