# Integration by Substitution

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Integration by Substitution

Section 6.2a

Slide2A change of variables can often turn anunfamiliar integral into one that we canevaluate…

This method is called thesubstitution method of integration.

The New Method

Slide3The New Method

Generally, this method is used when integrating a compositefunction, and the derivative of the inside function is alsopresent in the integrand:

1. Substitute

u

= g(x), du = g (x)dx

2. Evaluate by finding an

antiderivative

F (u) of f (u)

3. Replace

u

by

g

(

x

)

Slide4Initial Practice Problems

Evaluate

Let

Then

Substitute:

(Substitute)

Slide5Initial Practice Problems

Evaluate

Let

Then

Substitute:

Slide6Initial Practice Problems

Evaluate

Let

Then

Substitute:

Slide7Initial Practice Problems

Evaluate

Let

Then

Substitute:

Slide8Substitution in Definite Integrals

Instead of the last step we’ve been using (re-substitution???):

Substitute , and integrate with

respect to

u

from to .

Slide9Evaluating Definite Integrals

Evaluate

Let

Then

Also, notice:

Substitute:

Slide10Two Possible Methods?

Evaluate

Let

Then

Also, notice:

Method 1

Substitute:

Slide11Two Possible Methods?

Evaluate

Let

Then

Substitute:

Method

2

Slide12Guided Practice

Evaluate the given integral.

Slide13Guided Practice

Evaluate the given integral.

Slide14Guided Practice

Evaluate the given integral.

Slide15Guided Practice

Evaluate the given integral.

Slide16Guided Practice

Evaluate the given integral.

## Integration by Substitution

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