integrate sin ^-1[(x/a+x)^1/2]

Hi Suchita,

Make the substitution, x = a*tan²θ

So dx = 2a*tanθ*sec²θ dθ

and x/(a+x) = sin²θ

∫θ*2a*tanθ*sec²θ dθ = 2a∫θ secθ*(secθtanθ) dθ.

Integrate this by parts

= a*[θsec²θ - ∫sec²θdθ] = a[θsec²θ - tanθ] + constant.

where sec²θ = 1 + (x/a), and tanθ = √(x/a), θ = tan^-1√(x/a)

Hope that helps.

All the best,

Regards,

Ashwin (IIT Madras).