If \( \int \frac{\cos 4 x+1}{\cot x-\tan x} d x=A f(x)+B \), then \...

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If \( \int \frac{\cos 4 x+1}{\cot x-\tan x} d x=A f(x)+B \), then
\( \mathrm{P} \)
(1) \( A=-\frac{1}{8} \)
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(2) \( B=\frac{1}{2} \)
(3) \( f(x) \) has fundamental period \( \frac{\pi}{2} \)
(4) \( f(x) \) is an odd function
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