Is there an expression for a function that maps integers to one and non-integers to zero? Is there a function that can be built with addition, multiplication, exponentiation, trigonometric functions, integrals, (and all of their inverses i.e subtraction, division, taking logarithms, $\arcsin(x)$, derivatives, etc) that would take an integer and output $1$ and take a non-integer and output $0$?

I looked at the Dirichlet Function but it separated rationals from irrationals, and the only solution that I could come up with was with the Fourier Series of $$x-\lfloor x \rfloor = \frac 12 - \frac 1\pi \sum_{k=1}^\infty \frac 1k \sin (2 \pi k x)$$
But this uses infinite series. Is there a way to build such a function? If there is not, why not?
Maybe this is cheating, but I don't think it's possible otherwise, since all of the other functions are continuous and a composition of continuous functions is continuous, integrals preserve continuity too, and even though differentiation doesn't in general, all of the functions you listed are not just continuous but analytic or infinitely differentiable so their derivatives are continuous as well at all orders.  Anyway, here is my example:

$f(x) = 0^{\sin(\pi \cdot x)^2}$

For this to be correct you have to define $0^0=1$ which is not an unreasonable choice, but usually this expression is left undefined.
Well.. you can always built it as a distribution:

$$
\Upsilon(x) = 
\begin{cases}
1 ~~~ \text{for} ~~~ x\in\mathbb{N}
\\
0 ~~~ \text{for} ~~~x\not\in\mathbb{N}
\end{cases}
$$

lol