What is the integral of ${cos}^{2} (2 x) d x$ $cos^2(2x)dx$ ?

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What is the integral of ${cos}^{2} (2 x) d x$ $cos^2(2x)dx$ ?

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Answer 1 · 2017-01-15T03:53:35

$\int {cos}^{2} (2 x) d x = \frac{x}{2} + \frac{1}{8} sin (4 x) + C$ $int cos^2(2x)dx = x/2 + 1/8sin(4x) + C$

Explanation:

Use the identity:

${cos}^{2} θ = \frac{1 + cos 2 θ}{2}$ $cos^2theta= (1+cos2theta)/2$

so that:

$\int {cos}^{2} (2 x) d x = \int \frac{1 + cos (4 x)}{2} d x = \frac{1}{2} \int d x + \frac{1}{8} \int cos (4 x) d (4 x) = \frac{x}{2} + \frac{1}{8} sin (4 x) + C$ $int cos^2(2x)dx = int (1+cos(4x))/2dx = 1/2intdx + 1/8 int cos(4x)d(4x)= x/2 + 1/8sin(4x) + C$

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