Evaluate: `int1/xsqrt((1-sqrt(x))/(1+sqrt(x))dx)`

Explanation

Put ` x= cos^(2)theta`
`impliesdx= -2cos theta sin theta d theta`
` :. I=int(1)/(x)sqrt((1-sqrt(x))/(1+sqrt(x)))dx`
`=intsqrt((1-cos theta)/(1+cos theta))* (-2cos theta sin theta)/(cos^(2) theta) d theta`
`= -int ("sin"(theta)/(2))/("cos"(theta)/(2))*(4"sin"(theta)/(2)"cos"(theta)/(2))/("cos"theta)d theta `
`= -2 int (2 "sin"^(2)(theta)/(2))/(cos theta) d theta`
`= -2int(1-cos theta)/(cos theta)d theta`
`= -2 int (sec theta -1)d theta`
`= - 2 [log_(e)|sec theta+tan theta|-theta] +c`
`= -2[log_(e)|(1)/(sqrt(x))+(sqrt(1-x))/(sqrt(x))|-"cos"^(-1)sqrt(x)]+c`
`= -2[log_(e)|(1+sqrt(1-x))/(sqrt(x))|-cos^(-1)sqrt(x)] +c`