I try to solve for $a > 0$ the integral
$$\int_\mathbb{R} e^{ax} f(x) \,dx$$
where $f$ is the density of a $\mathcal{N}(0,1)$-distributed random variable (w.r.t. the Lebesgue measure) using the knowledge that for $Z \sim \mathcal{N}(0, \sigma^2)$ we have
$$\mathbb{E}[e^{Z}] = \exp(-\frac{\sigma^2}{2})$$
Sadly I can't see how to use this knowledge, since the factor $a$ can't be brought in front of the integral. Also I can't transform the random variable, since this would involve the density as well. What am I missing?
$$\int_{\mathcal{R}}e^{ax}f_X(x)dx=\int_{\mathcal{R}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(x^2-2ax+a^2)}e^{\frac{a^2}{2}}dx=$$
$$=e^{\frac{a^2}{2}}\int_{\mathcal{R}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(x^2-2ax+a^2)}dx=e^{\frac{a^2}{2}}\int_{\mathcal{R}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(x-a)^2}dx=e^{\frac{a^2}{2}}$$