Evaluate `int((1-sqrt(x))/(1+sqrt(x)))^(1//2)*(dx)/(x)`

Answer

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

Explanation

Let `I = int sqrt((1-sqrt(x))/(1+sqrt(x)))dx`
Put `x = cos^(2)theta rArr dx = -2 sin theta cos theta d theta`
`therefore I=intsqrt((1-cos theta)/(1+cos theta))*(-2 sin theta cos theta)d theta`
`=- int 2 "tan" (theta)/(2)*sin theta cos theta d theta =-2 int "sin"^(2)(theta)/(2)*cos theta d theta`
`=-2 int (1-cos theta)cos theta d theta =-2 int (cos theta-cos^(2)theta)d theta`
`=-2 int cos theta d theta + int (1+cos 2 theta)d theta`
`=-2 sin theta+theta+(sin 2 theta)/(2)+c`
`=-2 sqrt(1-x)+cos^(-1)sqrt(x)+sqrt(x(1-x))+c`