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Question
Integrate the function:
`tan^(-1) sqrt((1-x)/(1+x))`
Solution
Let `I = int tan^-1 sqrt ((1 - x)/(1 + x)) dx`
Let x = cos θ
⇒ dx = -sinθ dθ
`= I = int tan^-1 sqrt ((1 - cos theta)/(1 + cos theta)) - sin theta d theta`
`= int - tan^-1 (tan theta/2) (sin theta) d theta`
`= - int theta/2 sin theta d theta`
`= -1/2 [theta int sin theta d theta - int d/(d theta) (theta) int sin theta d theta] d theta`
`= -1/2 [theta (- cos theta) - int 1 (-cos theta) d theta]`
`= 1/2 theta cos theta - 1/2 int cos theta d theta`
`= 1/2theta cos theta - 1/2 sin theta + C`
`= 1/2 theta cos theta - 1/2 sqrt (1 - cos^2 theta) + C`
`= 1/2 [x cos^-1 x sqrt (1 - x^2)] + C`
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