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Antiderivatives and Improper Integrals, Lecture notes of Calculus

The concept of antiderivatives, their relationship with derivatives, and the fundamental property of antiderivatives. It also introduces the concept of improper integrals and provides examples of computing them with the constant of integration.

Typology: Lecture notes

2021/2022

Uploaded on 09/27/2022

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The Antiderivative
Taking an antiderivative involves doing the opposite of taking
a derivative.
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The Antiderivative

Taking an antiderivative involves doing the opposite of taking a derivative.

The Antiderivative

Taking an antiderivative involves doing the opposite of taking a derivative.

So, suppose that f (x) is a function. A function F (x) is called an antiderivative of f (x) if the derivative of F (x) is f (x).

So, we have

F ′(x) = f (x).

Examples:

  • An antiderivative of f (x) = 3x^2 − 2 x + 1 is

F (x) = x^3 − x^2 + x.

(Another antiderivative is F 1 (x) = x^3 − x^2 + x + 7.)

Example:

  • An antiderivative of f (x) = e^2 x^ is

Fundamental Property of Antiderivatives

Suppose that f (x) is a continuous function. If F (x) and G(x) are both antiderivatives of f (x) then there is some constant C so that

G(x) = F (x) + C.

Fundamental Property of Antiderivatives

Suppose that f (x) is a continuous function. If F (x) and G(x) are both antiderivatives of f (x) then there is some constant C so that

G(x) = F (x) + C.

On the other hand, if F (x) is an antiderivative of f (x) then for any constant C the function

G(x) = F (x) + C

is also an antiderivative of f (x).

Therefore, if we can find one antiderivative of f (x), then we know what all of the antiderivatives of f (x) are.

[WARNING: When doing an improper integral, you should al- ways write a constant of integration. Otherwise you will lose points on an exam.]

Summary:

The expression: (^) ∫

f (x)dx

represents an arbitrary antiderivative of f (x). If F (x) is any antiderivative of f (x) then we write ∫ f (x)dx = F (x) + C.