MAT 1235Calculus II
4.5 Part I
The Substitution Rule
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Homework
WebAssign HW 4.5 I There are 28 problems. Do it early. These problems ensure you to attain
certain degree of proficiency in this topic.
Preview
Antiderivatives are difficult to find. We need techniques to help us.
The substitution rule transforms a complicated integral into a easier integral.
The procedures for indefinite and definite integrals are similar but different.
Part I: Indefinite Part II Definite
Introductory Story
The wonderful design of windshield wipers
Introductory Story
The wonderful design of the integral notation…
The Substitution Rule for Indefinite Integrals
If is differentiable and is continuous on the range of , then
duufdxxgxgf )()())((
The Substitution Rule for Indefinite Integrals
If is differentiable and is continuous on the range of , then
duufdxxgxgf )()())((
dcomplicate easier
xin function uin function
Remarks
The key of the sub. rule is to find the sub. In practice, we do not memorize the formula The design of the integral notation
allows us to simplify the integral without using the formula (explicitly). For all practical purposes, we consider
dxxgdxxg )()(
Wonderful Design of Notation…
If , then...( )u g x
du
dxdu
( )( ( ))
( )
g x df g x
f u
x
du
Example 1
dxxx 42 )3(10
Example 1 dxxx 42 )3(10
2 3
2
2
u x
dux
dxdu xdx
du
4u
dxxx 42 )3(10
Analysis
Example 1 dxxx 42 )3(10
Cxdxxx 5242 )3()3(10
You can always check the answer by differentiation:
4252 )3(10)3( xxCxdx
d
Substitution Method
1.Select a substitution that appears to simplify the integrand. In particular, try to select so that is a factor in the integrand.
2.Express the integral entirely in terms of
and in one step.
3.Evaluate the new (and easier) integral.
4.Express the integral in terms of the original variable.
Expectations
Use a two-column format. Supporting info is on the right hand
column. Do not interrupt the flow of the main “solution line”.
Replace all the by in one step. Never have an integral with both variables.
Example 2
dxxx 12
Example 3
xdx2sin
Example 4
dttt 23)1cos(
Example 5
dt
t 63
4