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

Explanation

`" Let " I =int sqrt((1-sqrt(x))/(1+sqrt(x)))dx`
`"Let " sqrt(x) =cos t`
`rArr x=cos^(2) t rArr dx=2 cos t(-sint) dt`
`I=int sqrt((1-cos t)/(1+cos t)) (-2 sin t cos t) dt `
` I=-2 int sqrt((2sin^(2).(t)/(2))/(2cos^(2).(t)/(2))).2 sin.(t)/(2) cos .(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+ (1)/(2) sin 2t +C`
`=-2 sin t+t+(1)/(2) xx 2 sin t. cos t+C`
`=-2sqrt(1-cos^(2))t+1`
`+(1)/(2) (2sqrt(1-cos^(2))t.cost )+C`
`=-2sqrt(1-x)+cos^(-1) sqrt(x) +sqrt(x-x^(2))+C`