We notice that the integrand contains even power of
cosine, so the best strategy of evaluating the integral is to use the identity of half
angle.
(sin x)^2 = (1 - cos
2x)/2
We'll raise to square both
sides:
(sin x)^4 = (1 - cos
2x)^2/4
We'll integrate:
Int
(sin x)^4 dx = Int (1 - cos 2x)^2dx/4
We'll expand the
square:
Int (1 - cos 2x)^2dx/4 = (1/4)*Int (1 - 2cos 2x +
(cos 2x)^2)dx
Int (1 - cos 2x)^2dx/4 = x/4 - sin2x/4 +
(1/4)*Int (cos 2x)^2dx
Int (cos 2x)^2dx = Int (1+cos
4x)dx/2
Int (cos 2x)^2dx = x/2 + sin
4x/8
We'll multiply
by1/4:
(1/4)*Int (cos 2x)^2dx = x/8 + sin
4x/32
Int (sin x)^4 dx = x/4 - sin2x/4 + x/8 + sin 4x/32 +
C
Int (sin x)^4 dx = (1/4)*(3x/2 - sin 2x +
sin 4x/8) + C
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