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If \( \int f(x) \sin x \cos x d x=\frac{1}{2\left(b^{2}-a^{2}\right)} \log f(x)+c \) where \( c \) is constant of integration, then \( f(x)= \)
(A) \( \frac{2}{\left(b^{2}-a^{2}\right) \sin 2 x} \)
(B) \( \frac{2}{a b \sin 2 x} \)
(C) \( \frac{2}{\left(b^{2}-a^{2}\right) \cos 2 x} \)
(D) \( \frac{2}{a b \cos 2 x} \)
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