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 0(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 0(x) = G 0(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 of the form F (x) + C for some constant C.

D.L. White (Kent State University) 3 / 6 Formulas

Reversing any diﬀerentiation formula gives an antidiﬀerentiation formula.

For example, since d xm = mxm−1, dx it follows that the general antiderivative of f (x) = xn (for n 6= −1) is

1 F (x) = xn+1 + C. n + 1 Examples: 5 1 6 The general antiderivative of f (x) = x is F (x) = 6 x + C. −3 1 −2 The general antiderivative of f (x) = x is F (x) = − 2 x + C. √ 1/2 2 3/2 The general antiderivative of f (x) = x = x is F (x) = 3 x + C. 1 −7 1 −6 The general antiderivative of f (x) = x7 = x is F (x) = − 6 x + C.

D.L. White (Kent State University) 4 / 6 Formulas

Other basic antidiﬀerentiation formulas are the following (with F 0 = f and G 0 = g, and k a constant):

Function Antiderivative n 1 n+1 x (n 6= −1) n+1 x + 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) f (x) G(x) + C is not the antiderivative of g(x) .

D.L. White (Kent State University) 5 / 6 Examples

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

1 5 F (x) = 3 5 x + (− cos x) − 5x + C 3 5 = 5 x − cos x − 5x + C.

2 The general antiderivative of

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

2 4 7 3 10 2 F (x) = 4 x + 3 x + 2 x + 35x + C 1 4 7 3 2 = 2 x + 3 x + 5x + 35x + C.

1 3 2 (Note that this is not ( 3 x + 5x)(x + 7x) + C.)

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