Integration of Rational Functions

Definition A rational function is a quotient $\frac{P(x)}{Q(x)}$ of polynomials with $Q$ not the zero polynomial. It is proper if $\deg P < \deg Q$ and improper if $\deg P \ge \deg Q$.

Definition For polynomials $P$ and $Q$ with $Q$ not zero, there are unique polynomials $S$ and $R$ with

Then $\frac{P}{Q} = S + \frac{R}{Q}$ at every $x$ with $Q(x) \neq 0$, where $S$ is a polynomial and $\frac{R}{Q}$ is proper.

Definition Every nonconstant polynomial $Q$ with real coefficients factors into a constant times linear factors $(ax + b)$ with $a \neq 0$ and irreducible quadratic factors $(ax^2 + bx + c)$ with $a \neq 0$ and $b^2 - 4ac < 0$. If $\frac{P}{Q}$ is proper, it equals a sum of partial fractions, uniquely determined, made of the following terms. Each linear factor $(ax + b)$ appearing exactly to the power $r$ contributes

Each irreducible quadratic factor $(ax^2 + bx + c)$ appearing exactly to the power $s$ contributes

The $A_i$, $B_j$ and $C_j$ are real constants.

Definition To find the constants, multiply both sides of the decomposition by $Q(x)$. This gives an identity between two polynomials that holds for all $x$ (first for $x$ with $Q(x) \neq 0$, then for all $x$ by continuity). Two polynomials are equal for all $x$ exactly when the coefficients of each power of $x$ agree. Equating coefficients, or substituting convenient values of $x$ such as the roots of $Q$, gives linear equations for the constants.

Definition A linear factor $(ax + b)$ that appears to the power $r \ge 2$ needs all $r$ terms $\frac{A_k}{(ax + b)^k}$ for $k = 1, \ldots, r$. For $k \ge 2$ they integrate to powers,

Definition A quadratic $ax^2 + bx + c$ with real coefficients is irreducible over the reals when $b^2 - 4ac < 0$, so it has no real roots. Its partial fractions have linear numerators $Bx + C$. A term over the first power of the quadratic is integrated by splitting off a multiple of the derivative of the quadratic, which gives a logarithm, and completing the square in the rest, which gives an arctangent by

Definition An antiderivative of $\frac{P}{Q}$ found this way is valid on each open interval on which $Q$ has no zero. On different such intervals the constants of integration may differ. Suppose $\frac{P}{Q}$ is in lowest terms, so $P$ and $Q$ have no common factor. A definite integral $\int_a^b \frac{P(x)}{Q(x)} \, dx$ can be evaluated with the fundamental theorem only if $Q$ has no zero in $[a, b]$. Otherwise the integral is improper and needs separate treatment.