1. 6 Integration Antiderivatives and the Rules of Integration
Integration by Substitution
Evaluating Definite Integrals

2. (6.1): Antiderivative
A function F is an antiderivative of f on an interval I if
for all x in I.
Ex.
is an antiderivative of
Since

3. Theorem
Let G be an antiderivative of f. Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.
Notice
are all antiderivatives of

4. read “the integral of f,”
The process of finding all antiderivatives of a function is called integration.
The Notation:
read “the integral of f,”
means to find all the antiderivatives of f.
Indefinite integral
Integral sign
Integrand

5. Basic Rules
Rule
Example

6. Basic Rules
Rule
Ex. Find the indefinite integral

7. Initial Value Problem
To find a function F that satisfies the differential equation
and one or more initial conditions.
Ex. Find a function f if it is known that:
Gives C = 3

8. (6.2): Integration by Substitution
Method of integration related to chain rule differentiation.
Ex. Consider the integral:
Sub to get
Integrate
Back Substitute

9. Integration by Substitution
Steps:
1. Pick u = f (x), often the “inside function.”
2. Compute
3. Substitute to express the integral in terms of u.
4. Integrate the resulting integral.
5. Substitute to get the answer in terms of x.

10. Ex. Find
Pick u, compute du
Sub in
Integrate
Sub in

11. Ex. Find

12. Ex. Find

13. (6.4): Fundamental Theorem of Calculus
Let f be continuous on [a, b]. Then
where F is any antiderivative of f.

14. Evaluating the Definite Integral
Ex. Evaluate