Antidifferentiation

In this topic:

    (1) Define an antiderivative function.
    
    (2) If is an antiderivative of the continuous function then any other antiderivative, of must have the form where is some constant.
    
    (3) Find the family of antiderivatives of the function and write an equation using the indefinite integral notation.
    
    (4) Find the family of antiderivatives of the function , and write an equation using the indefinite integral notation.
    
    (5) Find the family of antiderivatives of the function and write an equation using the indefinite integral notation.
    

Definition (Antiderivative) A function is called an antiderivative of a given function on an interval if for all in

    The next theorem states that two antiderivatives of the same function differ only by a constant.

Theorem (Antiderivative) If is an antiderivative of the continuous function then any other antiderivative, of must have the form where is some constant.

    The notation where is an arbitrary constant means that is an antiderivative of It is called the indefinite integral of and satisfies the condition that for all in the domain of It is important to remember that represents a family of functions.

Example (Antiderivative) Find the family of antiderivatives of the function and write an equation using the indefinite integral notation.

    Solution.  If then and so an antiderivative of sine is By the Antiderivative Theorem, the most general antiderivative is Therefore, where is an arbitrary constant.

Example (Antiderivative) Find the family of antiderivatives of the function , and write an equation using the indefinite integral notation.

    Solution.  Since the general antiderivative of is , where is a constant, which is valid for because is defined on the interval Therefore, where is an arbitrary constant and

Example (Antiderivative) Find the family of antiderivatives of the function and write an equation using the indefinite integral notation.

    Solution.  Since   where is an arbitrary constant.

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Antidifferentiation
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/antidifferentiation.html