Question

Find the integrals of the function: $\frac {cos\ 2x}{(cos \ x+sin \ x)^2}$

Answer 1

$\frac {cos\ 2x}{(cos \ x+sin \ x)^2}$ = $\frac {cos\ 2x}{cos^2 x +sin^2 x+ 2sin \ x.cos\ x}$ = $\frac {cos\ 2x}{1+sin \ 2x}$

∴ ∫$$\frac {cos\ 2x}{(cos \ x+sin \ x)^2} dx = ∫$$\frac {cos\ 2x}{1+sin \ 2x}dx

Let 1+sin 2x = t

⇒ 2cos 2x dx = dt

∴ ∫$$\frac {cos\ 2x}{(cos \ x+sin \ x)^2} dx =$\frac 12 ∫\frac 1tdt$

=$\frac 12$ log|t| + C

= $\frac 12$log |1+sin 2x| + C

= $\frac 12$log |(sin x + cos x)2| + C

= log |sin x + cos x| + C