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

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

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Answer 1 · 2016-05-16T10:50:16

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

Explanation:

$cos (α + β) = cos (α) cos (β) - sin (α) sin (β)$ $cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)$
Making $α = β \to cos (2 α) = {cos (α)}^{2} - {sin (α)}^{2}$ $alpha=beta->cos(2alpha) = cos(alpha)^2-sin(alpha)^2$
but ${cos (α)}^{2} + {sin (α)}^{2} = 1$ $cos(alpha)^2+sin(alpha)^2=1$ then
${cos (α)}^{2} = \frac{1 + cos (2 α)}{2}$ $cos(alpha)^2=( 1+cos(2 alpha))/2$
so $\int {cos (x)}^{2} d x = \int \frac{1 + cos (2 x)}{2} d x = \frac{1}{2} \int d x + \frac{1}{2} \int cos (2 x) d x$ $int cos(x)^2dx = int( 1+cos(2 x))/2dx = 1/2int dx + 1/2intcos(2x)dx$
Finally $\int {cos (x)}^{2} d x = \frac{x}{2} + \frac{1}{4} sin (2 x) + C$ $int cos(x)^2 dx = x/2 + 1/4 Sin(2 x)+C$

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