Module 18  Antiderivatives as Indefinite Integrals and Differential Equations  
Introduction  Lesson 1  Lesson 2  Lesson 3  Lesson 4  SelfTest  
Lesson 18.1: Antiderivatives as Indefinite Integrals  
This lesson exlores the relationship between antiderivatives and indefinite integrals and discusses families of curves. Mathematics can be discovered using the TI89, as illustrated in Module 2 and Module 10. There is a sense of ownership and interest that is acquired with inductive learning. Review the discoverylearning process which was described in Module 2 and Module 10 and is shown below.
Defining Indefinite Integrals Recall that an antiderivative of a function f is a function F whose derivative is . The Fundamental theorem gives a relationship between an antiderivative F and the function f. , where F'(x) = f(x) and a is any constant. A modified notation is used to signify the antiderivatives of f. The new notation is called an indefinite integral and the antiderivatives of f are written as
Using the Integral Key The integral key, which is used to find definite integrals, can also be used to find indefinite integrals by simply omitting the limits of integration.
Exploring Examine the antiderivative of each of the following functions that have the form x^{n} and look for a pattern that will lead you to a general rule for finding .
Notice that the arbitrary constant C is not part of the result given by the TI89.
Describing the Pattern and Predicting
18.1.1 Describe the pattern you found when evaluating the indefinite integrals above and use it to predict
.
18.1.2 Confirm your prediction of
on your TI89. Extending the Examples Extend the exploration of examples by predicting the following indefinite integrals.
18.1.3 Confirm your predictions by entering the integrals on your TI89. Generalizing the Pattern
18.1.4 Predict a general rule for
and confirm it by entering the integral on your TI89. Checking Indefinite Integrals Because you can check each indefinite integral's result by finding the derivative of the result. For example, can be verified by taking the derivative of the result: . Because the result of the differentiation is the original function, the integration is confirmed. The Generalized Rule The generalized version of this rule is , where and C is a constant. Recall that the derivative of a constant is 0, so for any constant C. Illustrating
The indefinite integral
may be illustrated by graphing the family of curves that are represented by
for different values of C. For example,
corresponds to
. Letting C take the values 240, 200, 160, 120, 80, 40, 0, 40, 80, the family of curves is shown below in a [0, 50] x [50, 100] window where the particular curve associated with Each curve in the family can be obtained by choosing a different value of C and vertically translating the curve corresponding to C = 0. 

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