MAT 1221Survey of Calculus

Section 6.2The Substitution Rule

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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.