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If \( A \) is square matrix and \( e^{A} \) is defined as
P \( e^{A}=I+A+\frac{A^{2}}{2 !}+\frac{A^{3}}{3 !}+\cdots=\frac{1}{2}\left[\begin{array}{ll}f(x) & g(x) \\ g(x) & f(x)\end{array}\right] \), where

W \( A=\left[\begin{array}{ll}x & x \\ x & x\end{array}\right] \) and \( 0x1, I \) is an identity matrix.
\( \int \frac{f(x)}{\sqrt{g(x)}} d x \) is equal to
(1) \( \frac{1}{2 \sqrt{e^{x}-1}}-\operatorname{cosec}^{-1}\left(e^{x}\right)+c \)
(2) \( \frac{2}{\sqrt{e^{x}-e^{-x}}}-\sec ^{-1}\left(e^{x}\right)+c \)
(3) \( \frac{1}{2 \sqrt{e^{2 x}-1}}+\sec ^{-1}\left(e^{x}\right)+c \)
(4) none of these
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