Reference: G.H. Hardy - A Course of Pure Mathematics (ed. 3) page 229-230
I am working on the above book and am on the section where he introduces integrals. The way he begins to introduce them is to suppose there is a function $\psi(x)$. Then we wish to determine a function such that $\phi'(x)=\psi(x)$
He then provides his definition of the integral:
Definition: If $\psi(x)$ is the derivative of $\phi(x)$, then we call $\phi(x)$ an integral, or integral function of $\psi(x)$. The operation of forming $\psi(x)$ from $\phi(x)$ we call integration.
What I don't understand is why to go to the trouble of introducing $\psi(x)$. Instead of writing:
it would be just as valid to write:
Without having to make the step of introducing $\psi(x)$.
Question: Why does Hardy introduce another new function instead of just using the derivative function?