Problem: Find a uniformly valid approximation to the solution of $\epsilon y''(x) - (1 + \sin x)\, y'(x) - y(x) = 0$ with boundary conditions $y(0) = 1, y(1) = 1$ in the limit $\epsilon \to 0^+$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment.
Solution: $\boxed{y(x,\epsilon)=\exp\left( -\int_0^x \frac{dt}{1 + \sin t} \right)+\left(1 - 0.493\right) e^{-(1 + \sin 1)\, \frac{1 - x}{\epsilon}}}$