Question

Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`

Answer 1

Let I = `int cot^-1 ((1 + sinx)/cosx)*dx`

`(1 + sinx)/cosx = (1 + cos(pi/2 - x))/(sin(pi/2 - x)`

= `(2cos^2(pi/4 - x/2))/(2sin(pi/4 - x/2)*cos(pi/4 - x/2)`

= `cot(pi/6- x/2)`

∴ I = `int cot^-1 [cot(pi/4 - x/2)]*dx`

= `int (pi/4 - x/2)*dx`

= `pi/(4) int 1*dx - 1/2 int x*dx`

= `pi/(4)*x - (1)/(2)*x^2/(2) + c`

= `pi/(4)x - (1)/(4)x^2 + c`.