If $\int \frac{f\text{'}\left(x\right)x-f\left(x\right)}{\left(f\left(x\right)+x\right)\sqrt{f\left(x\right)x-x^2}}dx$  is equal to $\sqrt{m}{\tan }^{-1}\left(\sqrt{\frac{f\left(x\right)-x}{nx}}\right)+c,$  where m, n $\in$  R and ‘c’ is constant of integration (x > 0) then $\frac{m}{n}$ is equal to

We are tasked with evaluating the integral:

∫ [f'(x) * (x - f(x)) / (f(x) + x) * (xfxx - x²)] dx    

and determining the ratio m/n, where the solution is expressed as:

I = m * tan⁻¹(f(x) - x) / n * x + c,    

where m and n are real numbers, and c is the constant of integration.

Step-by-Step Solution

1. Rewrite the given integral in terms of its components:

   I = ∫ [f'(x) * (x - f(x)) / (xfxx)] dx            

2. Simplify the terms to make substitution feasible:

   I = ∫ [f'(x) * (x - f(x)) / (xfxx²)] dx.            

3. Let us substitute a new variable:

   Let t² = xfxx - x.                Therefore, f'(x) * (x - f(x)) / (xfxx²) dx = 2t dt.            

4. Substitute the simplified terms back into the integral:

   I = ∫ [2t / (t² + 2t)] dt.            

5. Integrate the simplified expression:

   I = 2 * tan⁻¹(t) + c.            

6. Replace t with its original value in terms of xfxx:

   t = sqrt(xfxx - x).            

   Hence,

   I = 2 * tan⁻¹((f(x) - x) / x) + c.            

7. Compare the final expression to the given form:

   I = m * tan⁻¹(f(x) - x) / n * x + c.            

   By comparison:

   m = 2, n = 2.            

Final Result

The ratio m/n is:

        m/n = 2/2 = 1.