In the Calculus book by Ron Larson, he presents 4 steps to construct partial fractions. The second step involves completely factoring the denominator into factors such that they are irreducible.
Now in the proof for the logistic function, we are trying to use partial fractions to integrate the term in the left of this equation.
Integral to solve using partial fractions
If I pretend that the denominator of the left term is already factored I can easily solve the integral using partial fractions. However, my question is how do I check that, indeed, the denominator term is factored.
HINT: we have $$\frac{1}{y\left(1-\frac{y}{L}\right)}=\frac{1}{y}-\frac{1}{y-L}$$