Partial fractions in integration

The integral of the polynomial function of the n^th degree

f(x)=a₀ + a₁ x + a₂ x² + ... + a_n xⁿ

is, by linearity of integration and by Formula 1) in Table of integrals:

F(x) = a₀ x + a₁ x²/2 + ... + a_n xⁿ⁺¹/(n+1) + C

A rational function f(x) = p(x)/q(x), p being of the m^th and q of the n^th degree, when m ≥ n can always be resolved into an polynomial function and a proper fractional function i.e. a function in which the degree of the numerator is at most n - 1.

This only requires the division by the denominator q(x) to be carried out until the order of the remainder becomes less than n. As the integration of the integer function has been given already, we have only to determine the integral of the form:

∫ p(x)/q(x) dx

in which p is of lower degree than q and they have no common root. Such an integral can sometimes be determined by using the natural logarithm integral condition.

This proper fractional function can be resolved into a sum of fractions with constant numerators and with denominators that are linear functions or powers of linear functions.

Let x₁, x₂, ..., x_n the n roots of q(x) be real or complex, but first let them all be different.

The proper fractional function can be resolved in only one way into partial fractions of the form:

<math>\frac{p \left( x \right)}{q \left( x \right)} = \frac{A_1}{x - x_1} + \frac{A_2}{x - x_2} + \cdots + \frac{A_n}{x - x_n}</math>

For multiplying both sides of the equation by q(x), we have:

<math>p \left( x \right) = \frac{A_1}{x - x_1} q \left( x \right) + \frac{A_2}{x - x_2} q \left( x \right) + \cdots + \frac{A_n}{x - x_n} q \left( x \right)</math>

Substituting for q(x) everywhere its value as the product (x - x₁) (x - x₂) ... (x - x_n) each denominator cancels; and if we put in for x any root x_k of q, all terms having the factor (x - x_k) disappear, leaving the term with the coefficient A_k; so that:

<math>p \left( x_k \right) = A_k \lim_{x = x_k}{\frac{q \left( x \right) }{x - x_k}} = A_k q' \left( x_k \right)</math>

thus

<math>A_k = \frac{p \left( x_k \right)}{q'\left( x_k \right)}</math>

q'(x_k) cannot vanish, since q(x) = 0 has only distinct roots. Obviously, since q'(x) would vanish for a multiple root of q(x) = 0, this method does not apply when there are multiple roots. Accordingly:

<math>\frac{p \left( x \right)}{q \left( x \right)} = \sum_{k = 1}^{n}{\frac{p \left( x_k \right)}{q'\left( x_k \right)} \frac{1}{x - x_k}}</math>

The final formula is

<math>\int{\frac{p \left( x \right)}{q \left( x \right)}\,dx} = \frac{p \left( x_1 \right)}{q'\left( x_1 \right)} \ln \left( x - x_1 \right) + \frac{p \left( x_2 \right)}{q'\left( x_2 \right)} \ln \left( x - x_2 \right) + \cdots + \frac{p \left( x_n \right)}{q'\left( x_n \right)} \ln \left( x - x_n \right) + C</math>

Need to cover case of multiple and/or complex conjugate roots.