- #1

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[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+u

^{2})

^{2}du................(2)

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

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

^{2}) - 2(1-a)[inte]du/(u

^{2}+1)

^{2}

The answer derived from the first part, (2-2a)[inte]du/(1+u

^{2}), contains tan

^{-1}but the model answer of this question is

-[squ](ax-x

^{2}) + 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?