Question

Find the integrals of the function:

`(cos 2x)/(cos x + sin x)^2`

Answer 1

Let `I = int (cos 2x)/((cos x + sin x)^2)`

`= int (cos^2 x - sin^2 x)/((cos x + sin x)^2)  dx`

`= int ((cos x - sin x) (cos x + sin x))/(cos x + sin x)^2  dx`

`= int (cos x - sin x)/(cos x + sin x)  dx`

Put `cos x + sin x = t`

`(- sin x + cos x) dx = dt`

`therefore I = int dt/t = log abs t + C`

`= log abs (cos x + sin x) + C`