Problem: Consider the following integral:$\int_0^{5} ( \frac{e^{-x}}{1 + x^2}) e^{-\epsilon (\frac{\sin^2(x)}{1 + x^4})} dx$In the limit$\epsilon \rightarrow \infty$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{\sqrt{\frac{2 \pi}{2 \epsilon}}}$