The Indefinite Integral

Explore:

Find a function with the given derivative.

1) f’(x) = 2x

2) y’ = 3x2

3) f’(x) = 2

1

x

Antiderivatives

x2 is an antiderivative of 2x.

Why “an”?

Because the antiderivative of 2x could also be x2 + 5

Let f(x) be a differential equation (a derivative)

Call F(x) an antiderivative of f(x)

Then G(x) will be the antiderivative where

G(x) = F(x) + C

Terms & Notation

G(x) = F(x) + C

Process is called antidifferentiation → indefinite integration

dxxfCxF )()(

general antiderivative antiderivative constant of integration

Integrand Variable of integration

integral

In book…

Know thepower rule!

)()(

)()(

xfdxxfdx

d

CxFdxxF

Integration is the “inverse” of differentiation

Differentiation is the “inverse” of integration

Example: Describe the antiderivatives of 2

4

x

Practice Time !!!

Just a Few More !!!

Initial Conditions & Particular Solutions

There are infinitely many solutions until you are told an initial condition about F(x) → F(2) = 5

Plug in the initial condition and solve for C

5 = (2)4 – (2)2 + C

C = -7

F(x) = x4 – x2 - 7

dxxxy 24 3 Cxx 24

F(x) → general solution

F(x) → particular solution