Hello aspirant,

Hope you are doing well!!

Integral of cos2x / ((cosx + sinx) power 2)

It is quite difficult to write here. But I'm trying my best.

there is no integration or power sign so we represent these sign with some other sign.

# represent the integration, and

** represent the power sign

when you can do this in notebook please change these sign with integration and power.

we know that,

cos2x / (cos x + sin x)**2 = cos2x / (cos**2x + sin **2x + 2sinxcosx

= cos2x / (1 + sin2x)

Now, # cos2x / (cos x +sin x)**2 dx

= # cos2x / (1 + sin2x) dx

Let, 1+ sin2x = t

2cos2xdx = dt

# (cos2x / (cos x + sin x)**2) dx

= 1/2 # 1/t dt

= 1/2 log |t| + C

= 1/2 log | 1 + sin2x| + C

= 1/2 log |(sin x + cos x )**2| + C

= log | sin x + cos x | + C

I hope you can understand.

All the best!!