Evaluate $\int_0^\infty f(x)dx$ when ,$f(x)=e^{-e^{x}}e^{x}\sin x$ by residues or otherwise

Question

Evaluate improper integral $\int_0^\infty f(x)dx$ . $$f(x)=e^{-e^{x}}e^{x}\sin x$$

My try- Can I have a semi circular path from $0$ to $R$ and counterclockwise semi-circle?

Answer 1

Your integral is \begin{align*} \int_0^{ + \infty } {e^{ - e^x } e^x \sin xdx} & = \Im \int_0^{ + \infty } {e^{ - e^x } e^x e^{ix} dx} = \Im \int_0^{ + \infty } {e^{ - e^x } e^{(1 + i)x} dx} \\ & \mathop = \limits^{t = e^x } \Im \int_1^{ + \infty } {e^{ - t} t^i dt} = \Im \Gamma (1 + i,1) = 0.1866485915\ldots, \end{align*} where $\Gamma(a,z)$ is the (upper) incomplete gamma function.