Calculus 6.2

1
Integration by Substitution
Greg Kelly Hanford High School Richland,
Washington
M.L.King Jr. Birthplace, Atlanta, GA
Photo by Vickie Kelly, 2002
2
The chain rule allows us to differentiate a wide
variety of functions, but we are able to find
antiderivatives for only a limited range of
functions. We can sometimes use substitution to
rewrite functions in a form that we can integrate.
3
Example 1
The variable of integration must match the
variable in the expression.
Dont forget to substitute the value for u back
into the problem!
4
Example
Solve for dx.
5
Example 3
6
Example
7
Example 7
8
Example
New Limits!
We could have substituted back and used the
original limits.
9
Example
Using the original limits
Wrong! The limits dont match!
10
Example
Dont forget to use the new limits.
11
In another generation or so, we might be able to
use the calculator to find all integrals.
Until then, remember that half the AP exam and
half the nations college professors do not allow
calculators.
You must practice finding integrals by hand until
you are good at it!
p