5.a – Antiderivatives and The Indefinite Integral.

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Presentation transcript:

5.a – Antiderivatives and The Indefinite Integral

Antiderivatives Definition A function F is called an antiderivative of f if F ′(x) = f (x) for all x on an interval I. Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is the family of functions given by F(x) + c, where c is an arbitrary constant. F(x) + c is a called a family of antiderivatives. If c is known, the antiderivative is called a specific antiderivative.

The Indefinite Integral Definition The general antiderivative, F(x), of a function f (x) can be represented by an indefinite integral Like the derivative, the dx denotes the variable with which we are anti-differentiating.

Examples Evaluate the indefinite integral (that is, determine the general anti-derivative) of the flowing functions. Use number three to develop a formula for

Properties of the Indefinite Integral Let c be a constant. Note: Always simplify the integrand before evaluating an integral.

Examples Evaluate the indefinite integral (that is, determine the general anti-derivative) of the flowing functions.

Examples Determine f if …

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Table of Basic Indefinite Integrals

Examples Evaluate the indefinite integrals.