Evaluate: `int(sin^(-1)sqrt(x)-cos^(-1)sqrt(x))/(sin^(-1)sqrt(x)+cos^(-1)sqrt(x))\ dx`

Answer

`(2)/(pi){sqrt(x-x^(2))-(1-2x)sin^(-1)sqrt(x)}-x+C`

Explanation

`I=int(sin^(-1)sqrt(x)-((pi)/(2)-sin^(-1)x))/((pi//2))dx=(2)/(pi)*int(2sin^(-1)sqrt(x)-(pi)/(2))dx`
`=(4)/(pi)intsin^(-1)sqrt(x)dx-intdx=(4)/(pi)*intsin^(-1)sqrt(x)dx-x+C`
`=(4)/(pi)[inttsin2tdt]-x+C," where "sin^(-1)sqrt(x)=t`.