Question

`int (cos 2x)/(sin x + cos x)^2dx` is equal to ______.

Options

• `(- 1)/(sin x + cos x) + "C"`

• log |sin x + cos x| + C

• log |sin x - cos x| + C

• `1/(sin x + cos x)^2`

Answer 1

`int (cos 2x)/(sin x + cos x)^2dx` is equal to log |sin x + cos x| + C.

Explanation:

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

`= 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

`= int dt/t = log t + C`

= log |sin x + cos x| + C