Here's the question that I got stuck:(adsbygoogle = window.adsbygoogle || []).push({});

[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?

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# Integration of irrational function

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