How to show $\int\frac{d}{dx}(a^u)dx=a^u+C$ more rigorously? We all know that $$\int\frac{d}{dx}(a^u)dx=a^u+C$$ where I am differentiating with respect to $x$. But how can I write it in a more rigorous way like for example using the Fundamental Theorem of Calculus or any other method?
How can I deduce that $$\int d(a^u)=a^u+C$$
The notation looks a bit odd for me because normally we view integral as the antiderivative or the limit of Riemann sum, but this time what does $\int d(a^u)$ actually mean?
Also, when are we allowed to swap the order of differentiation and integration?
Sorry, maybe I need to clarify that $a$ is a constant and $u$ is a function of $x$.
Just for reference, my knowledge about differentials is that $dy=f'(x)dx$.
Thanks for the explanation.

For the first one, just swap the order of integration and differentiation.
The latter follows from the definition of integration of a differential. Let $y = a^u$. Then integrate $dy$ and substitute $a^u$ for $y$ in the result of this integration.
