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Question
Integrate the function:
`sqrt((1-sqrtx)/(1+sqrtx))`
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Solution
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`
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