TITLE: $\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \mathrm dx$ Evaluate Integral
QUESTION [11 upvotes]: Evaluate
$$\int_{0}^{\infty} \frac{\cos x - e^{-x}}{x} \ dx$$

REPLY [3 votes]: $$\int_0^\infty  {{{\cos x - {e^{ - x}}} \over x}dx}  = \int_0^\infty  {\left( {{s \over {1 + {s^2}}} - {1 \over {1 + s}}} \right)ds}  = \mathop {\lim }\limits_{s \to \infty } \log {{\sqrt {{s^2} + 1} } \over {s + 1}} = 0$$