Let the solution curve \( y=y(x) \) of the differential equation \( \left(1+e^{2 x}\right)\left(...

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Let the solution curve \( y=y(x) \) of the differential equation \( \left(1+e^{2 x}\right)\left(\frac{d y}{d x}+y\right)=1 \) pass through the point \( (0, \pi / 2) \). Then, \( \lim _{x \rightarrow \infty} e^{x} y(x) \) is equal to
(a) \( \frac{\pi}{4} \)
(b) \( \frac{3 \pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{3 \pi}{2} \)
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