What is signified by the use of a "big" integral sign? [photo example]

Question

I encountered what I'll call, for lack of a better term, a "big" integral sign in a generic form of Integral Product Rule. I call it "big" relative to those preceding, and especially to the one immediately following it, in this example. This notation is strange (unfamiliar) to me. Please share your experience with this symbol.

Answer 1

It is the integration-by-parts formula, although demonstrated in a quite confusing way. Especially, it is unclear which function $\int$ is applied to. Using parentheses to emphasize the scope of $\int$ for each instance, we may instead write

$$\int(fg) = \left(\int f\right) g - \int \left(\left( \int f \right) g'\right). $$

It is still an unconventional way of demonstrating the IbP formula. So let me revert it to the good old way. Let $F $ denote an anti-derivative of $f$, i.e., $F' = f$. Then

$$ \int f(x)g(x) \, \mathrm{d}x = F(x)g(x) - \int F(x)g'(x) \, \mathrm{d}x. $$

Alternatively, it may help to adopt programmer-style notation for this. Write $\mathtt{Integrate}[f]$ for any ani-derivative of $f$ and $\mathtt{D}[g]$ for the derivative $g'$. Then the handwriting translates to

$$ \mathtt{Integrate}[f \cdot g] = \mathtt{Integrate}[f] \cdot g - \mathtt{Integrate}[\mathtt{Integrate}[f] \cdot \mathtt{D}[g]]. $$

Answer 2

He is writing an integral of a product, and the two factors are $\int f$ and $g'$. So he wrote the outside integral bigger.

$$ \int\;{\textstyle \int f} \cdot g' $$