Question

Evaluate:

`int tan^-1 sqrt((1 - x)/(1 + x)) dx`

Answer 1

Let `I = int tan^-1 sqrt((1 - x)/(1 + x)) dx`

Let x = cos t

⇒ dx = – sin t dt

∴ `I = int tan^-1 sqrt((1 - cos t)/(1 + cos t)) (-sin t  dt)`

= `int tan^-1 sqrt((2sin^2  t/2)/(2 cos^2  t/2)) (-sint) dt`

= `int tan^-1 (tan  t/2) (-sin t) dt`

= `-int t/2 sin t  dt`

= `-1/2 int t sin t  dt`

= `-1/2 [t(-cos t) - int 1. (- cos t) dt]`   ...[Integrating by parts]

= `1/2 t cos t - 1/2 sin t + c`

= `1/2 t cos t - 1/2 sqrt(1 - cos^2 t) + c`

= `1/2 x cos^-1 x - 1/2 sqrt(1 - x^2) + c`