why use a capital letter for the differential of an integral

Question

I have seen the notation for an integral of the form

$\int x dF(x)$, my professor told me that this is the same as $\int x f(x) dx$ Why introduce a different notations it it has the same meanings?

Answer 1

It's common to work with derivatives and antiderivates, so we use $F(x)$ and $f(x)$ to know these are related functions.

This is also common in probability where $f(x)$ is generally used to denote the probability density function, whereas $F(x)$ represents the associated cumulative distribution function.

Answer 2

To put @Bye_World's comment into an answer: the "antiderivative" of $f$ is something that $f$ is the derivative of:

$$ \begin{align} \frac{dF(x)}{dx} &= f(x) && \text{antiderivative}\\ dF(x) &= f(x)\,dx && \text{"multiplied" by $dx$}\\ \end{align} $$

Answer 3

For the same reason you're taught $a^2-b^2=(a+b)(a-b)$. Sure, they equate to the same thing, but they compute differently and as a result one may be easier to evaluate than the other. In your specific example, it equates functions like $\int x\ d(\cos x)$ and $\int x\ \sin x\ dx$.

As for the capital versus lowercase, that is a standard notation for "antiderivatives not expressed in integral form". Without that distinction, the Fundamental Theorems of Calculus, not to mention the equation your teacher gave, would be meaningless.