Evaluate : `int " sin"^(-1) sqrt((x)/(a+x)) dx `

Explanation

` int sin^(-1) sqrt((x)/(a+x)) dx`
` "Let " x= a tan^(2) 0`
` rArr" " dx =2a tan 0 . sec^(2) 0 d0`
`=int sin^(-1) sqrt((atan^(2) 0)/(a+a tan^(2) 0 )). 2a tan 0 sec^(2) 0d0`
`=2a intsin^(-1) sqrt((a tan^(2) 0)/(a sec^(2) 0)) . tan 0 sec^(2) 0d0`
`=2a int sin^(-1) (sin0) . tan 0 sec^(2) 0d0`
`=2aint0. (tan 0sec^(2)0) d0`
`=2a [0. int tan 0 sec^(2) 0 d0-`
`int {(d)/(d0) .0. int tan 0 sec^(2) 0 d0 } d0]`
` =2a [ 0. (tan^(2)0)/(2)- int 1. (tan^(2)0)/(2) d0]`
`=a[0. tan^(2)0-int (sec^(2) 0-1) d0]`
`=a[0. tan^(2) 0-tan 0+ 0] +c`
` =a[tan^(-1) sqrt((x)/(a)).(x)/(a)-sqrt((x)/(a))+ tan^(-1) sqrt((x)/(a))]+c`
` =(a +x) tan^(-1) sqrt((x)/(a)) - sqrt(ax) +c.`