The value of integral \( \mathrm{P}^{10} \) \( \int e^{x}\left[\fra...

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The value of integral
\( \mathrm{P}^{10} \) \( \int e^{x}\left[\frac{2 \tan x}{1+\tan x}+\cot ^{2}\left(x+\frac{\pi}{4}\right)\right] d x \) is equal to:

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(1) \( e^{x} \tan \left(\frac{\pi}{4}-x\right)+C \)
(2) \( e^{x} \tan \left(x-\frac{\pi}{4}\right)+C \)
(3) \( e^{x} \tan \left(\frac{3 \pi}{4}-x\right)+C \)
(4) \( e^{x} \tan \left(x-\frac{3 \pi}{4}\right)+C \)
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