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MAT 1221Survey of Calculus
Section 6.2The Substitution Rule
http://myhome.spu.edu/lauw
Today The rest of the quarter 6.2 Return exam 2 to you Please take advantage of all the bonus
points available to you!
What Does This picture Have To Do with Today’s Topic?
What Does This picture Have To Do with Today’s Topic?
𝑥5+𝑥3−75 𝑥4+3 𝑥2
What Does This picture Have To Do with Today’s Topic?
√𝑥5+𝑥3−75 𝑥4+3 𝑥2
Preview Antiderivatives are difficult to find. We
need techniques to help us. The substitution rule transforms a
complicated integral into a easier integral.
It is considered as the reverse process of the chain rule
The Smart Design of the Integral Notation
The differential encoded the information of the independent variable.
Placed at the right hand side to facilitate computations such as substitutions and integration by parts.
( )f x dx
The Substitution Rule for Indefinite IntegralsIf is differentiable and is continuous on the range of , then
duufdxxgxgf )()())((
The substitution Rule for Indefinite Integrals
duufdxxgxgf )()())((
dcomplicate easier
xin function uin function
If is differentiable and is continuous on the range of , then
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…
( )Let u g xdudxdu
( )( ( ))
( )
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 xdu xdxdu 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:
2 5( 3)d x Cdx
Substitution Method1.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 Let u
dudx
du
xdx
Bottom Line…
dxxx 12
There are not too many choices.
Example 3
22
3
5
x dxx
Let u
dudx
du
xdx
Example 4
dt
t 634 Let u
dudt
du
Expectations Follow the hints Change your variables in one single step If you are done early, do your HW. I will
wait for most of you to finish before returning the exam to you.