Integration of irrational function

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

Staff Emeritus
Science Advisor
Gold Member
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17
Check your work on your substitution.
 
u=sqrt[x/(a-x)]

dx = 2au/(1+u2)2 du

thanks
 

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