I wanted to know when you integrate sin^4x where the 4 comes from in the integral.
∫sin^4x dx= ∫(sin^2x)^2 dx= ∫(1-cos2x)^2/4 dx
cos(2x) = 2cos^2(x) - 1 = 1 - 2sin^2(x) = cos^2(x) - sin^2(x)
Well, for this problem, we are especially interested in the version containing only sin^2(x)
cos(2x) = 1 - 2sin^2(x)
Let's solve this for sin^2(x):
sin^2(x) = 1/2(1 - cos(2x))
Using this we can replace a sin^2(x) term (which we cannot integrate) with 1/2(1 - cos(2x)) (which we can integrate).
Thus the integration becomes:
∫sin^4x dx= ∫(sin^2x)^2 dx= ∫[1/2(1 - cos(2x))]^2 dx = ∫(1/2)^2*(1 - cos(2x)^2 dx = ∫(1-cos2x)^2/4 dx