Integration by Substitution

Lesson 5.5

Substitution with Indefinite Integration

• This is the “backwards” version of the chain rule

• Recall …

• Then …

5 42 24 7 5 4 7 2 4dx x x x x

dx

4 52 25 4 7 2 4 4 7x x x dx x x C

Substitution with Indefinite Integration

• In general we look at the f(x) and “split” it into a g(u) and a du/dx

• So that …

( )f x dx

( ) ( )du

f x g udx

( ) ( )du

g u dx G u Cdx

Substitution with Indefinite Integration

• Note the parts of the integral from our example

( ) ( )du

g u dx G u Cdx

4 52 25 4 7 2 4 4 7x x x dx x x C

Example

• Try this … what is the g(u)? what is the du/dx?

• We have a problem …

2(4 5)x dx

Where is the 4 which we need?

Where is the 4 which we need?

Example

• We can use one of the properties of integrals

• We will insert a factor of 4 inside and a factor of ¼ outside to balance the result

2)44

(41

5x dx

( ) ( )c f x dx c f x dx

31 1(4 5)

4 3x

Where did the 1/3 come from?

Where did the 1/3 come from? Why is this

now a 3?

Why is this now a 3?

Can You Tell?

• Which one needs substitution for integration?

• Go ahead and do the integration.

2

52

3 5

3 5

x x dx

x x dx

Try Another …

3 1t dt3sin cosx x dx

Assignment A

• Lesson 5.5

• Page 340

• Problems:1 – 33 EOO49 – 77 EOO

Change of Variables

• We completely rewrite the integral in terms of u and du

• Example:

• So u = 2x + 3 and du = 2x dx

• But we have an x in the integrand So we solve for x in terms of u

2 3x x dx

3

2

ux

Change of Variables

• We end up with

• It remains to distribute the and proceed with the integration

• Do not forget to "un-substitute"

1

21

2 3 32

x x dx u u du 1

2u

What About Definite Integrals

• Consider a variationof integral from previous slide

• One option is to change the limits u = 3t - 1 Then when t = 1, u = 2

when t = 2, u = 5 Resulting integral

5 1

2

2

1

3u du

2

1

3 1t dt

What About Definite Integrals

• Also possible to "un-substitute" and use the original limits

21 3 3

2 2 2

1

1 1 2 23 1

3 3 3 9u du u t

Integration of Even & Odd Functions

• Recall that for an even function The function is symmetric about the y-axis

• Thus

• An odd function has The function is symmetric about the orgin

• Thus

( ) ( )f x f x

0

( ) 2 ( )a a

a

f x dx f x dx

( ) ( )f x f x

( ) 0a

a

f x dx

Assignment B

• Lesson 5.5

• Page 341

• Problems:87 - 109 EOO117 – 132 EOO