is there any integral from zero to infinity that sums up to e?

Question

This may be very basic question, I just don't know. I just want to look and study that function. Thanks in advance

Edit: Sorry it may not meet requirements of a good question. I probably don't know what i need to know. You can delete

Answer 1

$$\int_0^\infty ce^{1-cx} \ dx, \text{ for each } c > 0$$

$$\int_0^\infty \frac {ce} {(x+c)^2} \ dx, \text{ for each } c > 0$$

$$\int_0^\infty c\chi_{[0,e/c]}(x) \ dx, \text{ for each } c > 0$$

where $\chi_A$ is the indicator function, $\chi_A(x) = \begin{cases} 1 & x \in A \\ 0, & x \not\in A \end{cases}$

etc.

Answer 2

$$f(x) = \left\{\begin{matrix}e, & 0 \le x \le 1 \\0, &\text{otherwise.}\end{matrix}\right.$$