Let \[ \begin{array}{l} \int \frac{f^{\prime}(x) g(x)-g^{\prime}(x) f(x)}{(f(x)+g(x)) \sqrt{f(x)...

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Let
\[
\begin{array}{l}
\int \frac{f^{\prime}(x) g(x)-g^{\prime}(x) f(x)}{(f(x)+g(x)) \sqrt{f(x) g(x)-g^{2}(x)}} d x \\
=\sqrt{m} \tan ^{-1}\left(\sqrt{\frac{f(x)-g(x)}{n g(x)}}\right)+C
\end{array}
\]
where \( m, n \in N \) and \( C \) is constant of integration \( (g(x)0) \). Find the value of \( \left(m^{2}+n^{2}\right) \).
(a) 2
(b) 4
(c) 6
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