integral sin^-1√{x/(a+x)}

put x=a tan²θ

dx=a 2tanθ sec²θ dθ .

then,

∫sin-1root(x/(a+x))

∫sin-1root(a tan²θ/(a+a tan²θ))a 2tanθ sec²θ dθ

∫sin-1root(tan²θ/sec²θ)2a tanθ sec²θ dθ

∫sin-1(tanθ/secθ)2a tanθ sec²θ dθ

∫sin-1(sinθ)2a tanθsec²θ dθ

2a∫θtanθ(1+tan²θ)dθ

2a[∫θ.tanθ dθ  + ∫θ.tan³θ dθ]

apply LIATE formula and solve it....