Unknown process in integration

Question

This is part of WolframAlpha’s calculations before creating partial fractions. What exactly is this method/theorem? And how does it work?

Answer 1

It’s not a method, it simply do this:

$$\int \frac 1{y(1-\frac yL)}dx= \int \frac 1{y\left(\frac{L-y}{L}\right)}dx=\int \frac 1{y\frac 1L(L-y)}dx= \int \frac 1{\frac 1L} \frac 1{y(L-y)}dx\underbrace=_{\frac 1{\frac 1L}=L} \int \frac L{y(L-y)}dx= \int \frac L{-y(y-L)}dx = \int -\frac L{y(y-L)}dx $$

Answer 2

It's algebra. $$\frac{1}{1 - \frac{y}{L}} \cdot \frac{L}{L} = \frac{L}{L-y}=-\frac{L}{y-L}$$ I suppose you could call it "multiplying by one"; it's a pretty common trick when working with fractions.

Calling it "canceling the common terms" is a little misleading, though. In this case, the common term is $\frac{1}{L}$; when you "cancel" that from $1$ you get $L$.