Why do some partial fractions have x or a variable in the numerator and others don't?

Question

Why do rational expressions like $\left(\frac{1}{(x-2)^3}\right)$ do not have x in the numerator of the partial fraction but a rational expression like $\left(\frac{1}{(x^2+2x+3)^2}\right)$ does have x in the numerator of its partial fraction?

Answer 1

You can integrate $\frac{1}{(x-2)^k}$ by itself, so it is not necessary to break it into the form $\frac{A}{x-2} + \frac{Bx+C}{(x-2)^2} + \dots.$ On the other hand, $\frac{1}{(x^2+2x+3)^k}$ does not have a simple antiderivative, so we must decompose it into the form $\frac{Ax+B}{x^2+2x+3} + \frac{Cx+D}{(x^2+2x+3)^2} + \dots$

The most general explanation is that $x^2+2x+3$ has complex roots while $x-2$ has real roots. If you allow complex numbers, you can break $\frac{1}{(x^2+2x+3)^k}$ into partial fractions where the numerator is always a constant instead of a term of the form $Ax+B.$