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

Answer

`2["cos"^(-1) sqrt(x) - "log"|1 + sqrt(1-x)|-(1)/(2)"log" |x|]+c`

Explanation

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