Really, integration is antiderivative, which means if you know how to do derivatives, you know how to do integrals. So, I'm not exactly sure what you mean by not using integrals. For example, this is an easy Calc 1 proof:
$\begin{align*}
f'(x) &= f(x) \\
\frac{f'(x)}{f(x)} &= 1 \\
\left(\ln (f(x))\right)' &= 1 \\
\ln (f(x)) &= x + C \\
f(x) &= e^{x + C} = a e^x
\end{align*}$
So, did I use integration? Or did I just have the knowledge that the only function whose derivative is 1 must be of the form $x + C$, for a constant $C$? All you need to know that is $\frac{d}{dx} x = 1$ and the following theorem:
If $F'(x) = G'(x)$ for all $x$ in $(a, b)$, then there is a constant $C$ such that $F(x) = G(x) + C$ for all $x$ in $(a, b)$.
This is given in the section on the Mean Value Theorem for Derivatives in Varburg, Purcell, and Rigdon, before antiderivatives or integrals are introduced. And, the proof, uses the MVT, and no theory on integration.