Integration Methods to Know for AP Calculus AB/BC

Antiderivative

The reverse process of differentiation, used to find integrals.

Integration by Parts

A technique using the product rule for differentiation, expressed as ∫u dv = uv - ∫v du.

Substitution

A method where part of the integrand is replaced with a new variable to simplify the integration.

Trigonometric Substitution

A technique used for integrating functions involving square roots of quadratic expressions using trigonometric functions.

Partial Fraction Decomposition

A method of simplifying rational functions by expressing them as sums of simpler fractions.

Improper Integral

An integral that has infinite limits of integration or an integrand that approaches infinity.

Trapezoidal Rule

A numerical integration method that approximates the area under a curve by breaking it into trapezoids.

Simpson’s Rule

A numerical method that approximates integrals using parabolas and is generally more accurate than the trapezoidal rule.

Basic Antiderivatives

The fundamental antiderivatives that include functions like x^n, e^x, sin x, and cos x.

Limit

A value that a function approaches as the input approaches some value.

Rational Function

A function that can be expressed as the quotient of two polynomials.

Composite Function

A function that is composed of two or more functions.

Integration Constant (C)

A constant added to the result of an indefinite integral.

Differentiation

The process of finding the derivative of a function.

Area Under Curve

The total space between the curve of a function and the x-axis, calculated using integrals.

Definite Integral

An integral that is evaluated over a specific interval, providing a numerical value.

Indefinite Integral

An integral without specified limits, representing a family of functions.

Continuous Function

A function without breaks, jumps, or interruptions in its graph.

Derivative

The rate at which a function is changing at any given point.

Integration Technique

A specific method used to perform integration, such as substitution or integration by parts.

Function Discontinuity

A point at which a function is not continuous, affecting integral evaluations.

Calculus

A branch of mathematics dealing with limits, functions, derivatives, integrals, and infinite series.

Analytical Solution

A solution that is expressed in exact mathematical terms rather than as an approximation.