Module 20 - Antiderivatives as Indefinite Integrals and Differential Equations | ||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self-Test | ||||||||||
Lesson 20.1: Antiderivatives as Indefinite Integrals | ||||||||||
This lesson explores the relationship between antiderivatives and indefinite integrals and discusses families of curves. You will also investigate finding antiderivatives graphically and analytically. Defining Indefinite Integrals
Recall that an antiderivative of a function f is a function F whose derivative is f, that is, ![]()
The Fundamental Theorem of Calculus gives another relationship between an antiderivative F and the function f: ![]() where F'(x) = f(x). If a is a constant, then -F(a) is a constant and we call it C, and the above can be written as ![]()
A modified notation,
![]() where F(x) is any one of the antiderivatives of f(x) and C is an arbitrary constant.
Exploring
Suppose you are to find an antiderivative of f(x) = x2. The Fundamental Theorem of Calculus says that by varying the value of a,
![]() This antiderivative looks like f(x) = x3. However, graphing Y2 = X^3 together with the antiderivative in Y1 shows they are not quite the same.
![]()
The antiderivative in the graph is actually
20.1.1 Graph an antiderivative of f(x) = x3 and find an equation for one antiderivative. Support your work graphically. Click here for the answer. Checking Indefinite Integrals Analytically
Because
![]() The Power Rule for Indefinite Integrals
The generalized version for antidifferentiation where
![]() Because the derivative of a constant is 0, the derivative of the result is ![]()
Illustrating
The indefinite integral
![]() 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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