My teachers shown an example of partial fraction in order to solve an integral. The problem is that I don't understand everything.
Say we have and that .
Then my notes say .
What I don't understand : why the numerators are not all constants, like a, b, c, etc. but appear to be lineal functions in the 3 last fractions?
I understood it better now. But I have a question :What about the example , can it be "reduced" to fractions with a quadratic function as numerator? I'm asking this because as you said, they are linear factors when the denominators are quadratic, so I bet that there are quadratic factors when the denominators are cubic. Am I right to bet so?They are linear factors when the denominators are quadratic.
The cubic can be factorised into a linear factor and an irreducible quadratic factor. So the denominator consists of a repeated linear factor and a repeated irreducible quadratic factor.
The cubic is difficult to factorise. And once factorised, the algebra in getting the partial factions is grinding and tedious.
So I'm afraid that you have two very unpleasant jobs if you want to express this rational function in partial fraction form.