Evaluating $\int_0^\infty e^{-x}\cos(x)dx$ By integrating over the contour around an appropriate sector, how does one solve  $$\int_0^\infty e^{-x}\cos(x)dx$$
Consider the integral
\begin{align}
\int_{C} e^{-|z|} \, e^{iz} dz
\end{align}
with a contour enclosing no poles.
$$\int_0^\infty f(x)dx=\lim_{x\rightarrow\infty}F(x)-F(0)$$