Glossary

Constant of Integration (+C)

Criticality: 3

An arbitrary constant added to the result of an indefinite integral. It represents the family of all possible antiderivatives of a function.

Example:

When integrating $f(x) = 2x$ , the result is $x^2 + C$ , where +C is the constant of integration.

Linear Factors

Criticality: 3

Factors of a polynomial that are of the first degree (e.g., $ax+b$). In partial fraction decomposition, the denominator is often factored into these terms.

Example:

For the denominator $x^2 - 5x + 6$ , the linear factors are $(x-2)$ and $(x-3)$ .

Long Division (for rational functions)

Criticality: 2

A polynomial division technique used when the degree of the numerator is greater than or equal to the degree of the denominator in a rational function. It simplifies the function into a polynomial plus a proper rational function.

Example:

If you need to integrate $\frac{x^3+1}{x-1}$ , you would first use long division to rewrite it as a polynomial plus a remainder term over $x-1$ .

Partial Fraction Decomposition (PFD)

Criticality: 3

A technique used to break down a complex rational function into a sum of simpler fractions, making it easier to integrate. It's particularly useful when the denominator can be factored into linear terms.

Example:

To integrate $\frac{5x-1}{(x-1)(x+4)}$ , you would use Partial Fraction Decomposition to rewrite it as $\frac{A}{x-1} + \frac{B}{x+4}$ .

Rational Function

Criticality: 2

A function that can be expressed as the ratio of two polynomials, where the denominator is not zero. Integration of these functions often requires specific techniques like partial fraction decomposition.

Example:

The function $f(x) = \frac{x^2 + 3x}{x^3 - 8}$ is a rational function because both the numerator and denominator are polynomials.

Undetermined Coefficients

Criticality: 3

A method used in partial fraction decomposition to find the unknown constants (A, B, C, etc.) in the numerators of the simpler fractions. This involves setting up and solving a system of equations.

Example:

After setting up $\frac{2x+1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$ , we use the method of undetermined coefficients by substituting specific x-values to solve for A and B.