Problem: Consider the following integral:$I(x) = \int_0^1[\frac{e^{-xt}}{1+t^2}]dt$In the limit$x \rightarrow 0$, find approximate behavior of the integral up to and including the leading order in x. Provide your answer in a $\boxed{}$ latex environment.
Solution: $\boxed{I(x) = \frac{\pi}{4}-\frac{x}{2}\ln(2)}$