Question

Please integrate it do it fast ​

Answer 1

- cosec x + cot x + x + c

Explanation:

`\int (cos x)/(1 + cos x) dx\\</p><p>=\int (cos x(1-cosx))/((1 + cos x)(1-cosx)) dx\\</p><p>=\int (cos x-cos^2 x)/((1-cos^2 x)) dx\\</p><p>=\int (cos x-cos^2 x)/(sin^2 x) dx\\</p><p>=\int \bigg ((cos x)/(sin^2 x)- (cos^2 x)/(sin^2 x)\bigg) dx\\</p><p>=\int (cotx \: cosecx- cot^2 x) dx\\</p><p>=\int [cotx \: cosecx- (cosec^2 x-1)] dx\\</p><p>=\int (cotx \: cosecx- cosec^2 x+1) dx\\</p><p>=\int (cotx \: cosec) dx- \int cosec^2 x dx+\int 1 dx\\</p><p>= - cosec x + cot x + </p><p>x + c`