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