Question

Integrate the function:

`sqrt((1-sqrtx)/(1+sqrtx))`

Answer 1

Let `I = sqrt((1 - sqrtx)/(1 + sqrtx))`dx

`sqrtx = cos t`

=> x = cos² t

dx = - 2 cos t sin t dt

I = `int sqrt((1 - cos t)/(1 + cos t)) * (- 2 cos t sin t) dt`

`= - 2 int sqrt((2 sin^2  t/2)/(2 cos^2  t/2)) cos t sin t dt`   ...`[because 1 + cos A = 2 cos^2  A/2, 1 - cos A = 2 sin^2  A/2]`

`= - 2 int (sin t/2)/(cot t/2) (2 sin  t/2 cos  t/2 t) dt    .....[because sin A = 2 sin  A/2  cos  A/2]`

`= - 4 int sin^2  t/2 cos t dt`

`= - 4 int (1 - cos t)/2 cos t  dt`

`= - 2 int (cos t - cos^2 t)  dt`

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

`= - 2 sin t + (t + (sin 2t)/2) + C`

`= - int (2 cos t - 1 - cos 2t)  dt`

`= - [2 sint - t - (sin 2t)/2] + C`

`= - [2 sin t - t - sin t cos t] + C`

`= - [2 sqrt (1 - x) -  cos ^-1 sqrt x - sqrt (1 - x) * sqrt x] + C`

`= -2 sqrt (1 - x) + cos^-1 sqrt x + sqrtx * sqrt (1 - x) + C`