Antiderivatives and Improper Integrals, Lecture notes of Calculus

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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.