I am reading through Finite Calculus: A Tutorial for Solving Nasty Sums by David Gleich and on page 9 he computes the general derivative of an exponent:
$$ \triangle(c^x) = c^{x+1} - c^x = (c-1)c^x $$
This part is clear to me. However then the text goes on to say
Because $c$ is a constant in this expression, we can then immediately compute the anti-derivative as well
$$ \sum (c^x)\delta x = \frac{c^x}{c-1} + C $$
How is this immediate computation done? Are the rules of infinite calculus being applied?
The idea is that the sum and the difference cancel each other, hence
$$c^x=\sum\triangle(c^x) \delta x= \sum (c-1)c^x\delta x=(c-1)\sum c^x\delta x,$$
which results in
$$\sum (c^x)\delta x = \frac{c^x}{c-1} + C.$$
We need to add the constant because functions differing in a constant have the same difference.