Week 02-1: Partial Fraction Decomposition
We will take a look at how to take integrals of any rational function by expressing it as a sum of simpler fractions.
Partial Fraction Decomposition
Let’s demonstrate this with a rational function of the form $$f(x)=\frac{P(x)}{Q(x)}$$
where both P and Q are polynomials and the degree of \(P\) is less than the degree of \(Q\). This type of rational function is called proper. It is important to note that partial fractions can only be used if the degree of the numerator is strictly less than the degree of the denominator.
Recall (Degree of Polynomials)
If we have a polynomial of the form $$P(x)=a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$$
where \(a_n \neq 0\), then the degree of \(P\) is \(n\).
In the event that we do end up with a rational function where the degree of the numerator is greater than or equal to the degree of the denominator, we can use long division until a remainder \(R(x)\) is obtained where the degree of \(R\) is less than the degree of \(Q\) (the denominator). This division statement is $$ f(x) = \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)} $$
where \(S\) and \(R\) are also polynomials.
Factoring Term Table
Once we have determined that partial fractions can be used, we then use the following table to determine the decomposed terms:
| Denominator Factor | Partial Fraction Decomposition Term | |
|---|---|---|
| Case 1 (Linear Factors) | $$ ax+b $$ | $$ \frac{A}{ax+b} $$ |
| Case 2 (Repeated Linear Factors) | $$ (ax+b)^k $$ | $$ \frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \dots + \frac{A_k}{(ax+b)^k}, k \in \Z^+ $$ |
| Case 3 (Irreducible Quadratic Factors) | $$ ax^2+bx+c $$ | $$ \frac{Ax+B}{ax^2+bx+c} $$ |
| Case 4 (Repeated Irreducible Quadratic Factors) | $$ (ax^2+bx+c)^k $$ | $$ \frac{A_1x + B_1}{ax^2+bx+c} + \frac{A_2x + B_2}{(ax^2+bx+c)^2} + \dots + \frac{A_kx+B_k}{(ax^2+bx+c)^k}, k \in \Z^+ $$ |