Substitution

Lesson 7.2

Review

Recall the chain rule for derivatives

We can use the concept in reverse• To find the antiderivatives or integrals of

complicated formulas

We look for integrands that fit the right side of the chain rule

2

( ( )) '( ( )) '( )df g x f g x g x

dx

434 xx e dx 4210 1x x dx

Strategy

We look for an expression that can be the "inside" function

We substitute u = g(x)• We also determine what is du or g'(x)

3

( ( )) '( ( )) '( )df g x f g x g x

dx

4210 1x x dx

2 1 2 2du

u x x or du x dxdx

Integration by Substitution

Now we have

Then we use the general power rule for integrals

Finally substituteu = x2 + 1 back in 4

42 410 1 5x x dx u du

1

1

nn uu du C

u

4 55u du u C 52 1x C

Substitution Method

We seek the following situations where we can substitute u in as the "inner" functionLet u represent the quantity under a root or raised to a power

Let u represent the exponent on e

Let u represent the quantity in the denominator

5

1 x dx434 xx e dx

2

32 1

xdx

x

Example

Consider the problem of taking the integral of

Strategy … substitute u = 4x – 6• What is the derivative of u with respect to x?

Now we make the substitution• The ¼ adjusts for

the 4 in the du

6

74 6x dx

4

4

du

dxdu dx

71

4u du

Substitution

The resulting integral is much simpler

Now we reverse the substitution and simplify

7

7 81 1 1

4 4 8u du u C

814 6

32x C

Try Another

What will we substitute … u = ?

What is the du ?

Now rewrite the integral and proceed

8

42 3x x dx 2 3u x

2du x dx

54 5 21 1 1 13

2 2 5 10u dx u C x C

How About Another?

Consider u = ? du = ?

u = x2 + 5 du = 2x dx

Problem … • 2x is not a constant• Cannot adjust the integral with a constant

coefficient

Substitution will not work for this integral9

2 5

dx

x

Indefinite Integral of u -1

If it looked like thiswe could do it

u = x2 + 5 du = 2x dx

Then use rule for integral of u -1

Final result:

10

2

2

5

x dx

x

1 lnu du u C 2ln 5x C

Indefinite Integral of eu

Try this:

What is the u? the du?• u = x4 du = 4x3 dx

Rewrite, adjust for the factor of 4 in the du

11

43 xx e dx

1

4ue du

1

4ue C

41

4xe C

Practice

Try these

12

2

3

6

2 7

xdx

x

2 2r r dr2

yedyy

Application

We are told that a certain bacteria population is increasing a rate of

What is the increase in the population during the first 8 hours

13

1

2

1800'( )

18 0.5P t

t

Assignment

Lesson 7.2A

Page 449

Exercises 3 – 41 odd

Lesson 7.2B

Page 450

Exercises 39 – 44 all

14