Integration Using Partial Fractions 47

1. Core Concept Overview & Intuition

Integration by partial fractions is a technique for integrating a rational function, that is, a quotient of two polynomials. The central idea is to decompose a complicated rational integrand into a sum of simpler fractions whose integrals are already known from the basic formulae and the extension formulae. Once the decomposition is complete, each simpler fraction is integrated separately and the results are added.

The method rests on the algebraic fact that every polynomial with real coefficients can be factored into a product of linear factors and irreducible quadratic factors. The form of the partial-fraction decomposition is dictated entirely by the nature of the factors appearing in the denominator. Three structural cases arise in this lesson:

• The denominator factors into distinct (non-repeated) linear factors.
• The denominator contains an irreducible quadratic factor.
• The integrand is an improper fraction, meaning the degree of the numerator is greater than or equal to the degree of the denominator.

In the first two cases the integrand is a proper fraction, so the decomposition proceeds directly. In the third case a preliminary polynomial long division is required to separate a polynomial part from a proper fractional remainder, after which the proper fraction is decomposed as before.

The intuition is that integration is the reverse of differentiation, and the derivative of a logarithm produces a fraction of the form $\frac{1}{ax+b}$. Consequently, every non-repeated linear factor in a denominator contributes a logarithmic term to the integral. An irreducible quadratic factor, by contrast, contributes a logarithm together with an arctangent term, because the derivative of $\arctan x$ produces a denominator of the form $x^{2}+a^{2}$. Recognizing which factor type is present therefore predicts the shape of the final answer before any algebra is performed.

2. Mathematical Formalization & Core Definitions

> Definition (Rational Function). A rational function is a function of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero polynomial.

> Definition (Proper and Improper Fraction). The rational function $\frac{P(x)}{Q(x)}$ is called a proper fraction if the degree of $P(x)$ is strictly less than the degree of $Q(x)$. It is called an improper fraction if the degree of $P(x)$ is greater than or equal to the degree of $Q(x)$.

> Definition (Partial Fraction Decomposition). A partial fraction decomposition of a proper rational function $\frac{P(x)}{Q(x)}$ is an identity expressing it as a finite sum of simpler fractions whose denominators are the factors of $Q(x)$.

The decomposition rules for the factor types treated in this lesson are as follows.

> Rule (Distinct Linear Factors). If $Q(x)$ contains distinct linear factors $(a_{1}x+b_{1}), (a_{2}x+b_{2}), \ldots, (a_{n}x+b_{n})$, then there exist constants $A_{1}, A_{2}, \ldots, A_{n}$ such that

$$
\frac{P(x)}{Q(x)}=\frac{A_{1}}{a_{1}x+b_{1}}+\frac{A_{2}}{a_{2}x+b_{2}}+\cdots+\frac{A_{n}}{a_{n}x+b_{n}}.
$$

> Rule (Irreducible Quadratic Factor). If $Q(x)$ contains an irreducible quadratic factor $ax^{2}+bx+c$, then the corresponding term in the decomposition has the form $\frac{Bx+C}{ax^{2}+bx+c}$, where $B$ and $C$ are constants to be determined.

> Rule (Improper Fraction). If $\frac{P(x)}{Q(x)}$ is improper, first divide $P(x)$ by $Q(x)$ to obtain

$$
\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)},
$$

> where $S(x)$ is a polynomial and $\frac{R(x)}{Q(x)}$ is a proper fraction. The polynomial $S(x)$ is integrated using the basic power formula, and the proper fraction is decomposed by the preceding rules.

The constants in a decomposition are determined by clearing denominators and equating coefficients of like powers of $x$, or by substituting convenient values of $x$. The resulting integrals rely on the basic formulae already established, in particular

$$
\int \frac{1}{x}\,dx=\ln\lvert x\rvert+c,
$$

and, for a linear factor with non-unit leading coefficient, the extension formula

$$
\int \frac{1}{ax+b}\,dx=\frac{1}{a}\ln\lvert ax+b\rvert+c.
$$

For an irreducible quadratic factor, completing the square reduces the quadratic to the form $u^{2}+a^{2}$, and the standard result

$$
\int \frac{1}{u^{2}+a^{2}}\,du=\frac{1}{a}\arctan\frac{u}{a}+c
$$

supplies the arctangent contribution. Here $u$ denotes the completed-square variable and $a$ is a positive constant.