MATH 12002 - CALCULUS I 3.7: Antiderivatives (Part 1) Professor - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §3.7: Antiderivatives (Part 1)

Professor Donald L. White

Department of Mathematical Sciences Kent State University

D.L. White (Kent State University) 1 / 6

Definitions and Theorems

Definition

Let f be a function defined on an interval I. An antiderivative of f is a function F such that F ′(x) = f (x) for all x in I. For example, if f (x) = 3x2, then an antiderivative for f is F(x) = x3. But so are x3 + 5, x3 − 17, and x3 + 78.34. Since the derivative of a constant is 0, x3 + C is an antiderivative of 3x2 for any constant C.

D.L. White (Kent State University) 2 / 6

Definitions and Theorems

Recall one of the consequences of the Mean Value Theorem:

Theorem

If F(x), G(x) are functions such that F ′(x) = G ′(x) on an interval I, then G(x) = F(x) + C on I for some constant C; that is, F and G differ only by a constant on I. In other words, every antiderivative of 3x2 is x3 + C for some constant C. In general, we have

Theorem

If F is an antiderivative of f , then the general antiderivative of f is F(x) + C for an arbitrary constant C; that is, every antiderivative of f is

• f the form F(x) + C for some constant C.

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Formulas

Reversing any differentiation formula gives an antidifferentiation formula. For example, since d dx xm = mxm−1, it follows that the general antiderivative of f (x) = xn (for n = −1) is F(x) = 1 n + 1xn+1 + C. Examples: The general antiderivative of f (x) = x5 is F(x) = 1

6x6 + C.

The general antiderivative of f (x) = x−3 is F(x) = −1

2x−2 + C.

The general antiderivative of f (x) = √x = x1/2 is F(x) = 2

3x3/2 + C.

The general antiderivative of f (x) = 1

x7 = x−7 is F(x) = −1 6x−6 + C.

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Formulas

Other basic antidifferentiation formulas are the following (with F ′ = f and G ′ = g, and k a constant): Function Antiderivative xn (n = −1)

1 n+1xn+1 + C

sin x − cos x + C cos x sin x + C kf (x) kF(x) + C f (x) + g(x) F(x) + G(x) + C From our differentiation formulas, we also know that the general antiderivative of sec2 x is tan x + C and the general antiderivative of sec x tan x is sec x + C. Note that F(x)G(x) + C is not the antiderivative of f (x)g(x) and

F(x) G(x) + C is not the antiderivative of f (x) g(x).

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Examples

1 The general antiderivative of f (x) = 3x4 + sin x − 5 is

F(x) = 3 1

5x5

+ (− cos x) − 5x + C =

3 5x5 − cos x − 5x + C.

2 The general antiderivative of

f (x) = (x2 + 5)(2x + 7) = 2x3 + 7x2 + 10x + 35 is F(x) =

2 4x4 + 7 3x3 + 10 2 x2 + 35x + C

=

1 2x4 + 7 3x3 + 5x2 + 35x + C.

(Note that this is not (1

3x3 + 5x)(x2 + 7x) + C.)

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