# integration of irrational function

by KLscilevothma
Tags: function, integration, irrational
 P: 321 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?