What is the strategy used to evaluate the trigonometric integral of the function y=sin^4x?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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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