Question

What is the integral of `cos^2(x)`?

Answer 1

`int cos(x)^2 dx = x/2 + 1/4 Sin(2 x)+C`

Explanation:

`cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)`
Making `alpha=beta->cos(2alpha) = cos(alpha)^2-sin(alpha)^2`
but `cos(alpha)^2+sin(alpha)^2=1` then
`cos(alpha)^2=( 1+cos(2 alpha))/2`
so `int cos(x)^2dx = int( 1+cos(2 x))/2dx = 1/2int dx + 1/2intcos(2x)dx`
Finally `int cos(x)^2 dx = x/2 + 1/4 Sin(2 x)+C`