Today, I was explaining integration by parts to a fellow student, a freshman.
I ended up demonstrating how to calculate $\int e^{2x}\sin(3x)\,dx$ by iterative integration. At the end, I ended up with the equality
$$\frac{13}{4}\int e^{2x}\cos(3x)\,dx = \frac{1}{2}e^{2x}\sin(3x)-\frac{3}{4}e^{2x}\cos(3x)$$
and so
$$\int e^{2x}\cos(3x)\,dx = \frac{4}{13}\left(\frac{1}{2}e^{2x}\sin(3x)-\frac{3}{4}e^{2x}\cos(3x)\right).$$
This made me question what $\int f(x)\,dx$ even means to begin with. My textbook states that $\int f(x)\,dx$ denotes any of the primitive functions to $f(x)$. But if so, I've just showed that all of the primitive functions to $f(x)$ are equal to the right hand side of the second equation. This can obviously not be the case, as I could just add on any non-zero constant to obtain a new primitive.
So I guess my question is: what is the formal definition of the symbol $\int f(x)\,dx$?
EDIT: Turns out I was simply forgetting to add constants, as per Thomas' comment below. $\int f(x)\,dx$ does indeed denote an arbitrary primitive to $f$ when $f$ is defined on some interval.