This is a beta course, so its structure, chapters, and examples may continue to change.

Integrals of Rational Functions

Rational functions are integrated via partial fraction decomposition.

Partial Fraction Decomposition

Idea: decompose a complex rational function into simpler partial fractions.

Example 1 $\int \dfrac{1}{x^2 - 1} dx$

Factor $(x-1)(x+1)$ , set $\dfrac{1}{x^2-1} = \dfrac{A}{x-1} + \dfrac{B}{x+1}$ ; solve $A=\tfrac12$ , $B=-\tfrac12$ .

Result: $\tfrac12\ln|x-1| - \tfrac12\ln|x+1| + C$ .

Compute $\int \dfrac{1}{x^2 - 4} dx$ .

Reference Answer

partial fractionsrational function

Approach: Factor the denominator as $x^2-4=(x-2)(x+2)$ and decompose the integrand into partial fractions before integrating term by term.

Steps:

Factor: $x^2-4=(x-2)(x+2)$ .
Decompose: $\dfrac{1}{x^2-4} = \dfrac{1}{(x-2)(x+2)} = \dfrac{1/4}{x-2} - \dfrac{1/4}{x+2}$ .
Integrate each term: $\tfrac14\ln|x-2| - \tfrac14\ln|x+2| + C$ .

Answer: $\dfrac{1}{4}\ln|x-2| - \dfrac{1}{4}\ln|x+2| + C$ .

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\int$	mathematical symbol	integral	Indefinite/definite integral sign
$dx$	mathematical symbol	differential of x	The differential element of the integration variable $x$
$C$	constant	constant of integration	The arbitrary constant in every antiderivative
$x^2 - 1$	mathematical symbol	x squared minus one	The denominator that is factored before the decomposition
$(x-1)(x+1)$	mathematical symbol	the product of x minus one and x plus one	The factorisation of the denominator into linear factors
$A, B$	constant	A, B	The coefficients solved for when the partial fractions are matched
$\dfrac{A}{x-1} + \dfrac{B}{x+1}$	mathematical symbol	A over x minus one, plus B over x plus one	The pair of simple fractions the integrand is rewritten as
$\ln\lvert x-1\rvert$	mathematical symbol	natural logarithm of the absolute value of x minus one	The antiderivative of the first simple fraction
$\ln\lvert x+1\rvert$	mathematical symbol	natural logarithm of the absolute value of x plus one	The antiderivative of the second simple fraction
$x^2 - 4$	mathematical symbol	x squared minus four	The denominator factored in the exercise

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