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