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If \( A \) is square matrix and \( e^{A} \) is defined as \( e^{A}=I+A+\frac{A^{2}}{2 !}+\frac{A^{3}}{3 !}+\ldots=\frac{1}{2}\left[\begin{array}{ll}f(x) & g(x) \\ g(x) & f(x)\end{array}\right] \), where
\( \mathrm{W} \) \( A=\left[\begin{array}{ll}x & x \\ x & x\end{array}\right] \) and \( 0x1, I \) is an identity matrix.
\( \int(g(x)+1) \sin x d x \) is equal to
(1) \( \frac{e^{x}}{2}(\sin x-\cos x) \)
(2) \( \frac{e^{2 x}}{5}(2 \sin x-\cos x) \)
(3) \( \frac{e^{x}}{5}(\sin 2 x-\cos 2 x) \)
(4) none of these
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