integration of irrational function

KLscilevothma

Here's the question that I got stuck:

[inte]sqrt[x/(a-x)] dx ......................................(*)

I tried to use the following substitution
u=sqrt[x/(a-x)] and .........................................(1)
dx = 2u(1-a)/(1+u²)² du................(2)

sub (1) and (2) into (*), after a few steps, I got

(2-2a)[inte]du/(1+u²) - 2(1-a)[inte]du/(u²+1)²

The answer derived from the first part, (2-2a)[inte]du/(1+u²), contains tan ^-1 but the model answer of this question is
-[squ](ax-x²) + a/2sin^-1[(2x+a)/a] + C
For the second part, I let u = tan θ and got a strange expression.

Is my approach correct and is the final answer obtained from the above method differs the model answer only by the constant of integration ? Or am I using a wrong substitution?

Hurkyl

Check your work on your substitution.

KLscilevothma

u=sqrt[x/(a-x)]

dx = 2au/(1+u²)² du

thanks