Let be $S_n:[a,b]\to\mathbb{R}$ a differentiable function for all $n\in\mathbb{N}$ and $(S_n)_{n\in\mathbb{N}}$ a function series that converges for at least one point $c\in[a,b]$. Further, we assume that that the function series of $(S_n)_{n\in\mathbb{N}}$ derivatives, $(S'_n)_{n\in\mathbb{N}}$ converges uniformly.
Then we know that $(S_n)_{n\in\mathbb{N}}$ also converges uniformly, $\lim\limits_{n\to\infty}S'_n$ is differentiable and $\lim\limits_{n\to\infty}S'_n= (\lim\limits_{n\to\infty}S_n)'$.
(see https://en.wikipedia.org/wiki/Uniform_convergence#To_differentiability)
Now let's assume we want to find a closed form of $\lim\limits_{n\to\infty}S_n$ but the partial sums $S_n$ are somewhat complicated and it is hard to figure out how $\lim\limits_{n\to\infty}S_n$ looks like. However, we know that $\lim\limits_{n\to\infty}S'_n$ attains a simple form which allows us to easily find the antiderivative of $\lim\limits_{n\to\infty}S'_n$.
My question is, though we are able to find the antiderivative of $\lim\limits_{n\to\infty}S'_n$ it's not unique, so how do I know which of those antiderivatives is my desired expression for $\lim\limits_{n\to\infty}S_n$? Or am I mistaken and this strategy is not applicable in general?
An antiderivative of $\lim S_n'$ isn't unique but the limit of $(S_n)$ is because for some $c \in [a,b]$ the series of function converges at point $c$. Here, $\lim S_n$ is the antiderivative of $\lim S_n'$ taking the value $\lim S_n(c)$ at $c$ : $$ (\lim S_n) (x) = \lim (S_n(c)) + \int_c^x (\lim S_n')(t)dt. $$ Also, denoting $$ S_n = \sum_{k=0}^n f_k $$ this last formula becomes clearer : $$ \sum_{n=0}^{+ \infty} f_n(x) = \sum_{n=0}^{+ \infty} f_n(c) + \int_c^x \sum_{n=0}^{+ \infty} f_n'(t)dt$$ where one can swap the series and the integral. This strategy is very useful when dealing with a series of the kind you described : the series of the derivatives is nice but the series itself isn't.