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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?

pie
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log2
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  • @Mariano I presume this refers to integrating over cycles in an algebraic curve or variety? – Ted Shifrin Dec 30 '23 at 01:20
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    A famous example, Laplace integral: $$ \int_0^{\infty} \frac{\cos bx}{a^2+x^2}, \mathrm{d}x = \frac{\pi}{2a} \mathrm{e}^{-ab}, $$ where $a,b>0$. Proof can be found here. – Huanyu Shi Jan 01 '24 at 05:09
  • But in fact, trigonometric functions and exponential functions have a very profound connection. I'm not sure if this meets the OP's criteria – Huanyu Shi Jan 01 '24 at 05:22

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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 :

enter image description here

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

pie
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  • @MarianoSuárez-Álvarez I don't think it is possible to have $e$ without $\sin x , \sinh x$ etc – pie Dec 30 '23 at 00:56
  • @MarianoSuárez-Álvarez Well the question is not clear enough whether trig function are allowed or not. I found tons of examples of integral of trig function that have in the result of integration.. – pie Dec 30 '23 at 01:02
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    Nice answer. Another example is this integral. – log2 Dec 31 '23 at 08:22