Question

Evaluate: `int (sec x)/(1 + cosec x) dx`

Answer 1

`I = int (sec x)/(1 + cosec x) dx`

`I = int  (1/cos x)/((sin x + 1)/sin x) dx`

`I = int  1/(cos x) xx (sinx)/(1+ sin x) dx`

`I = int (tan x)/(1+sin x) dx`

`I = int (tan x)/(cos^2 x) (1- sin x)dx`

Put tan x = t

`sec^2 x dx = dt`

`= intt (1 - t/(sqrt(1+t^2)))dt`

`= int t dt - int  t^2/sqrt(1+t^2) dt` 

Let `I _1 = int t^2/sqrt(1 + t^2) dt`

`= t^2/2 - int [(1 + t^2 -1)/sqrt(1 + t^2)]dt`

`I_1 = int [sqrt(1+t^2) - 1/sqrt(1 + t^2)] dt`

`I_1 = 1/2 tsqrt(1 + t^2) + 1/2 log |t + sqrt(1 + t^2)| - log |t + sqrt(1 + t^2)| + c`

`I_1 = 1/2 tan x sqrt(1 + tan^2x) + 1/2  log |tan x + sqrt(1+tan^2 x)| - log |tan x + sec x|+ c`

`:. I = t^2/2 - [1/2 tan x . sec x + 1/2 log |tanx + sec x| - log |tanx + sec x|  + C]`

`:. I = (tan^2x)/2 - 1/2 tan x . sec x +  1/2  log |tan x +  sec x| + c`