Integral of Cosine

$\begin{aligned} I & = \int_0^\infty \frac {\cos x}{2^x} dx \\ & = \int_0^\infty \frac {\cos (-x)}{2^x} dx \\ & = \int_0^\infty \frac {\Re \left( e^{-ix}\right)}{2^x} dx \\ & = \Re \left( \int_0^\infty 2^{-x} e^{-ix} \ dx \right) \\ & = \Re \left( \int_0^\infty e^{-(i+\ln 2)x} \ dx \right) \\ & = \Re \left( - \frac {e^{-(i+\ln 2)x}}{i+\ln 2} \bigg|_0^\infty \right) \\ & = \Re \left(\frac 1{i+\ln 2} \right) \\ & = \Re \left(\frac {\ln 2 - i}{1+(\ln 2)^2} \right) \\ & = \boxed{\dfrac {\ln 2}{1+(\ln 2)^2}} \end{aligned}$