Question

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

Options

• `(-1)/(sin x + cos x)^2`

• `log |sin x + cos x| + c`

• `log |sin x - cos x| + c`

• `1/(sin x + cos x)^2`

Answer 1

`log |sin x + cos x| + c`

Explanation:

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

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

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

Put `cos x + sin x = t, (- sin x + cos x) = dt`

 = `int (dt)/t = log |t| + c`

= `log |sin x + cos x| + c`