Anti Differentiation

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

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.

Here are some basic antidifferentiation formulas followed by some examples on how to use them.

Constant Multiple:    

Sum Rule:         

Difference Rule:     

Linearity Rule:     

Constant Rule:     

Exponential Rule:     

Power Rule:         

Trigonometric Rules:













Inverse Trigonometric Rules:

Example (Antiderivative Formulas) Find the most general antiderivative of the function

    Solution. We want to evaluate Since, we find,
    

Example (Antiderivative Formulas) Find the most general antiderivative of the function

    Solution. We want to evaluate Since,



we use the linearity rule and the power rule to find,
    

Example (Antiderivative Formulas) Find given

    Solution. First we find by,
    


where is some constant which can be determined given So,



So in fact, Now to find we follow the same procedure,




where is a constant which can be determined given So,



Therefore, as desired.

Example (Antidifferentiation Application) Finding the demand function given the marginal revenue. A manufacturer estimates that the marginal revenue of a certain commodity is when units are produced. Find the demand function

    Solution. Since,
    


and because where is the demand function, we must have so that yielding and

Theorem (Area as an Antiderivative)  If is a continuous function such that for all on the closed interval then the area bounded by the curve the -axis, and the vertical lines and viewed as a function of is an antiderivative of on

Example (Area as an Antiderivative) Find the area under the parabola over the interval

    Solution. The area function is given by and we can determine using and so which means Therefore, the area under the curve from is

Example (Antidifferentiation Application) A ball is thrown upward with a speed of 48 ft/s from the edge of a cliff 432 feet above the ground. Find its height above the ground seconds later. When does it reach its maximum height? When does it hit the ground?

    Solution. The motion is vertical and we choose the positive direction to be upward. At time the distance above the ground is and the velocity is decreasing. Therefore the acceleration must be negative Taking the antiderivative, To determine we use the given information of Thus, and so and Therefore, the ball reaches it maximum height at which means Since and using we find that and so Therefore the height function is and so the ball hits the ground when meaning by using the quadratic formula.

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