In "A Quotient Rule Integration by Parts Formula", the authoress integrates the product rule of differentiation and gets the known formula for integration by parts: \begin{equation}\int f(x)g'(x)dx=f(x)g(x)-\int f'(x)g(x)dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\end{equation} This formula is for integrating a product of two functions. It can be named therefore product rule integration by parts formula.
Furthermore, the authoress integrates the quotient rule of differentiation and gets a formula which is for integrating a quotient of two functions:
\begin{equation}\int\frac{f'(x)}{g(x)}dx=\frac{f(x)}{g(x)}+\int \frac{f(x)g'(x)}{g(x)^{2}}dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\end{equation}
She names it therefore quotient rule integration by parts formula.
"Quotient-Rule-Integration-by-Parts" goes also not beyond this.
To get a real product rule for integration and a real quotient rule for integration, I substitute in equation $(1)$: $g'(x)\rightarrow g(x)$, and in equation $(2)$: $f'(x)\rightarrow f(x)$. For abbreviation, I set $F(x)=\int f(x)dx+c_{1}$ and $G(x)=\int g(x)dx+c_{2}$, where $c_{1}$ and $c_{2}$ are constants. So I get the following product rule of integration:
\begin{equation}\int f(x)g(x)dx=f(x)G(x)-\int f'(x)G(x)dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\end{equation}
and the following quotient rule of integration:
\begin{equation}\int\frac{f(x)}{g(x)}dx=\frac{F(x)}{g(x)}+\int\frac{F(x)g'(x)}{g(x)^{2}}dx\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\end{equation}
Formulas $(3)$ and $(4)$ bring nothing new. They seem to be more complex than the formula of integration by parts, and it is harder to remember this formulas than the formula of integration by parts. But this formulas are more clear for the inexperienced because he sees what he wants: a product or a quotient of two functions at the left side of the equations.
My questions to formulas $(3)$ and $(4)$ are:
Are this formulas already known?
What are the reasons why this formulas are not used?
