Calculus: Integration of Rational Functions Practice Flashcards

These flashcards cover key terminology and methods for integrating rational functions, including long division, partial fraction decomposition for linear, repeated, and irreducible quadratic factors, and completing the square.

Rational Function

A function that can be expressed as $P/Q$ where $P$ and $Q$ are polynomials.

Long Division

A method used when the degree of the numerator is greater than or equal to the degree of the denominator, allowing the expression to be rewritten as a quotient plus the remainder over the divisor.

Partial Fraction Decomposition

A technique used to break a rational function into simpler pieces when the degree of the denominator is larger than the degree of the numerator.

Difference of Squares

A factoring pattern used for denominators like $x^2 - 4$ to rewrite them as linear factors like $(x + 2)(x - 2)$.

System of Equations

A set of linear equations used during partial fraction decomposition to solve for unknown coefficients such as $A$ and $B$ by grouping terms or plugging in specific values.

Repeated Factor

A denominator factor that appears multiple times (e.g., $(2x + 1)^2$), requiring a separate partial fraction for every power of the factor according to the number of times it is repeated.

Irreducible Quadratic Factor

A quadratic factor in the denominator that does not factor into real linear terms, requiring a linear numerator in the form of $Ax + B$ in its partial fraction.

Completing the Square

A process used to rewrite a quadratic expression like $x^2 + 2x + 4$ into a perfect square form like $(x + 1)^2 + 3$ to facilitate integration.

Arc Tangent Integration Rule

An elementary anti-derivative formula where the integral of $\frac{1}{x^2 + 1}$ is equal to $\tan^{-1}(x)$ or $\text{arctan}(x) + C$.

Natural Logarithm Integration Rule

The anti-derivative of the reciprocal function $\frac{1}{u}$, which results in $\text{ln}(|u|) + C$.

U-Substitution (u-sub)

An integration technique used to simplify rational functions or trigonometric integrals by replacing a variable and its derivative, such as setting $u = \text{sin}(x)$ when its derivative $\text{cos}(x)$ is present in the numerator.