L ectur e 9| 1 Chapter 3
Integration
Outline 5. Substitution rule (Technique of Integration)
6. Definite integral using the substitution rule
L ectur e 9| 2 Substitution Rule
Let be a continuous function and
is a differentiable function. Then
where the integral on the right is taken with respect to , and then is setting back with .
L ectur e 9| 3 Proof Let . So is an antiderivative of , that is
To prove the substitution rule, we have to show that
which is the same as showing that
By the chain rule, we have
But , hence
as desired.
L ectur e 9| 4 EX (Look for a composite function)
Evaluate the integral
Remark There are 3 steps:
Step 1. Look for a composite function .
Step 2. Let and integrate
Step 3. Set back into Step 2.
L ectur e 9| 5 EX Evaluate
L ectur e 9| 6 The differential point of view
The substitution rule for
can be understood as substituting into the integral to get
L ectur e 9| 7 EX Evaluate the integral
L ectur e 9| 8 EX Evaluate the integral
L ectur e 9| 9 EX (Some prelim. algebra: Splitting)
Evaluate
L ectur e 9| 10
General Substitution Rule
For general integral of the form
it is a good guess to introduce
Then substitute
to get
To be success, one should be able to get rid of in using and replace by the variable !!!
L ectur e 9| 11 EX Evaluate the integral
L ectur e 9| 12 EX Evaluate
L ectur e 9| 13 EX Calculate
L ectur e 9| 14 EX Evaluate
L ectur e 9| 15 Substitution Rule: Definite Integrals
We have
For the general case, use the fundamental theorem of calculus #2.
Proof Follows from the fundamental theorem of calculus #2: If then
L ectur e 9| 16 EX Evaluate the definite integral
L ectur e 9| 17 EX Evaluate
L ectur e 9| 18 EX Evaluate
L ectur e 9| 19 EX Let be continuous. Show that
If, in addition, and , calculate
L ectur e 9| 20 Symmetry
Let be continuous.
1. If is even ( for all ), then
2. If is odd ( for all ), then
L ectur e 9| 21 Proof We can express
Use the substitution rule with , , and at , :
so
1. If is even then and
L ectur e 9| 22 2. If is odd then and
So the identities are proved.
L ectur e 9| 23 EX Evaluate the integral
