How do you find the antiderivative of $(cos(2x))^2$ ?

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Answer 1 · 2016-12-30T16:06:24

$F(x) = -x/2-1/8sin4x+C$

First we note that:

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

So, its primitive can be computed as:

$int_0^x cos^2(2t)dt = 1/2int_0^x (cos(4t)-1)dt = -x/2 + 1/8int_0^x cos(4t)d(4t) = -x/2-1/8sin4x+C$

How do you find the antiderivative of (cos(2x))^2(cos(2x))^2?