Answer:
- cosec x + cot x + x + c
Step-by-step explanation:
[tex] \int \frac{cos x}{1 + cos x} dx\\
=\int \frac{cos x(1-cosx)}{(1 + cos x)(1-cosx)} dx\\
=\int \frac{cos x-cos^2 x}{(1-cos^2 x)} dx\\
=\int \frac{cos x-cos^2 x}{sin^2 x} dx\\
=\int \bigg (\frac{cos x}{sin^2 x}- \frac{cos^2 x}{sin^2 x}\bigg) dx\\
=\int (cotx \: cosecx- cot^2 x) dx\\
=\int [cotx \: cosecx- (cosec^2 x-1)] dx\\
=\int (cotx \: cosecx- cosec^2 x+1) dx\\
=\int (cotx \: cosec) dx- \int cosec^2 x dx+\int 1 dx\\
= - cosec x + cot x +
x + c[/tex]
