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We know that the derivative of some non-elementary functions can be expressed in elementary functions. For example $ frac{d}{dx} Si(x)= frac{sin(x)}{x} $

So similarly are there any non-elementary functions whose integrals can be expressed in elementary functions?

If not then how can we prove that any integral in the set of non-elementary integrals cannot be expressed in the form of elementary functions?

The derivative of an elementary function is an elementary function: the standard Calculus 1 differentiation methods can be used to find this derivative. So an antiderivative of a non-elementary function can't be elementary.

EDIT: More formally, by definition an elementary function is obtained from
complex constants and the variable $x$ by a finite number of steps of the following forms:

1. If $f_1$ and $f_2$ are elementary functions, then $f_1 + f_2$, $f_1 f_2$ and (if $f_2 ne 0$) $f_1/f_2$ are elementary.


2. If $P$ is a non-constant polynomial whose coefficients are elementary functions, then a function $f$ such that $P(f) = 0$ is an elementary function.


3. If $g$ is an elementary function, then a function $f$ such that $f' = g' f$ or $f' = g'/g$ is elementary (this is how $e^g$ and $log g$ are elementary).

To prove that the derivative of an elementary function, you can use induction on the number of these steps. In the induction step, suppose
the result is true for elementary functions obtained in at most $n$ steps.
If $f$ can be obtained in $n+1$ steps, the last being $f = f_1 + f_2$ where $f_1$ and $f_2$ each require at most $n$ steps, then $f' = f_1' + f_2'$ where $f_1'$ and $f_2'$ are elementary, and therefore $f'$ is elementary. Similarly for the other possibilities for the last step.

No, the derivative of an elementary function is elementary; some integrals were defined specifically as the antiderivative of certain functions because that function otherwise would have no closed-form antiderivative.

An anti-derivative of a non-elementary function cannot be an elementary function.

Yes, and I can provide a simple counter-example.

Let $f(x)$ be piece-wise defined such that $f(x) = x^2$ for $x neq 0$ and such that $f(0) = 300$.

This is not an elementary function. However its integral is $F(x) = frac {1}{3}x^3 + c$ which is elementary.

For a slightly more "non-elementary" example just make $f(x) = -500$ whenever $x$ is an integer multiple of $n = 0.0001$. Feel free to keep decreasing $n$ to make the function messier and messier.

However, if you want a continuous non-elementary $f$ then no. If $f$ is continuous then by one of the fundamental theorems of calculus $F'(x) = f(x)$ and the derivative of an elementary function is an elementary function. Furthermore, if you want that $f$ is an integral of some other $h$ then it follows that $f$ is continuous as the integral of any real valued function defined everywhere is a continuous function. So this will only work with discontinuous $f$'s that are not integrals of other functions.

In short the set of derivatives of elementary functions $neq$ the set of anti-integrals of elementary functions.