Revealing $e$ in real integrals without $e$ or $e^x$ in the integrand

Question

I've observed that certain real integrals yields elements not present within the integrated function itself. For instance, integrating $\frac{1}{x}$ produces $\ln$, and specific definite integrals of $\frac{1}{x^2+1}$ involve $\pi$. Then, I'm seeking a definite or indefinite integral whose result contains $e$, despite the absence of $e$ or $e^x$ in the integrated function. Could you provide such an example or why it might not be possible?

Answer 1

I found this integral from this Telegram channel

for $b \in \mathbb{R}$ $$\int_0^{\infty}\cos\bigg(x \sqrt{x^2+b^2}\bigg)dx = \frac{ \sqrt{2\pi}}{4} e^{-\frac{b^2}{2}} $$

---

Proof :

There are many such examples in this channel it took me under a minute to find this.