Evaluate: `intsin^(-1)sqrt(x/(a+x))dxdot`

Explanation

`I=int sin^(-1)sqrt((x)/(a+x))dx`
Let `x=a tan^(2) theta " or " dx=2a tan theta sec^(2)theta d theta`
`:. intsin^(-1)sqrt(sin^(2)theta)2a sec^(2)theta tan theta d theta`
`=2a int underset(I)(underbrace(theta)) underset(II)(underbrace(sec^(2) theta tan theta)) d theta`
`=2a[theta(tan^(2)theta)/(2)-int(tan^(2)theta)/(2) d theta]`
`=a[ theta tan^(2)theta-int(sec^(2) theta-1) d theta]`
`=a(theta tan^(2) theta-tan theta + theta]+c, " where " theta ="tan"^(-1)sqrt((x)/(a))`