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If f'(x)=sin^2(2x), how to find the function f(x)?

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If f'(x)=sin^2(2x), how to find the function f(x)?

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To determine the primitive function f(x), we'll have to evaluate the indefinite integral of the function (sin 2x)^2.

We'll apply the half angle identity:

(sin 2x)^2 = (1 - cos 4x)/2

We'll evaluate the indefinite integral:

Int (sin 2x)^2dx = Int (1 - cos 4x)dx/2

We'll apply the property of integral to be additive:

Int (1 - cos 4x)dx/2 = Int dx/2 - Int cos 4x dx/2

Int (1 - cos 4x)dx/2 = x/2 - sin 4x/8 + C

The primitive of the given function is: f(x)=x/2 - sin 4x/8 + C.

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