MATH 12002 - Calculus I: Antiderivatives (Part 1), Lecture notes of Calculus

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MATH 12002 - CALCULUS I

§3.7: Antiderivatives (Part 1)

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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) = 3x^2 , then an antiderivative for f is F (x) = x^3. But so are x^3 + 5, x^3 − 17, and x^3 + 78.34.

Since the derivative of a constant is 0, x^3 + C is an antiderivative of 3x^2 for any constant C.

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 6 = −1) is

F (x) =

n + 1

xn+1^ + C.

Examples: The general antiderivative of f (x) = x^5 is F (x) = 16 x^6 + C. The general antiderivative of f (x) = x−^3 is F (x) = − 12 x−^2 + C. The general antiderivative of f (x) =

x = x^1 /^2 is F (x) = 23 x^3 /^2 + C. The general antiderivative of f (x) = (^) x^17 = x−^7 is F (x) = − 16 x−^6 + C.

Formulas

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

Function Antiderivative xn^ (n 6 = −1) (^) n+1^1 xn+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 sec^2 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).