INTEGRATION BY SUBSTITUTION

INTEGRATION BY SUBSTITUTION

Section 4.5Section 4.5

When you are done with your homework, you should be able

to…

– Use pattern recognition to find an indefinite integral

– Use a change of variables to find an indefinite integral

– Use the General Power Rule for Integration to find an indefinite integral

– Use a change of variables to evaluate a definite integral

– Evaluate a definite integral involving an even or odd function

Emilie du Châtelet lived from 1706-1749. She was a French mathematician. Though she

conquered the heart of Voltaire, she later fell in love with the Marquis de Saint-Lambert, a

courtier and very minor poet. She died several days after giving birth to his child. Which of the

following statements are true?

A. She explained one part of Leibnitz’s system in a book entitled Institutions de physique.

B. She translated Newton's Principia into French.C. She frequently claimed that the only pleasures

left for a woman when she is old is study, gambling, and greed.

D. All of the above.

Theorem: Antidifferentiation of a Composite Function

• Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, then

 • If , then and  

•

f g x g x dx F g x C u g x du g x dx

f u du F u C

PATTERN RECOGNITION

We need to recognize and

f g x

.g x

OutsideFunction

Derivative ofInsi

Inside Functio

de Functi n

n

o

dx F g x Cg xf g x

Which expression represents in

the integral shown?

A.

B.

C.

2 33 1x x dx g x

3 1x

3 1x

23x

Which expression represents in

the integral shown?

A.

B.

C.

2 33 1x x dx f x

3 1x

x

23x

Guidelines for Making a Change of Variables

1.Choose a substitution . Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power or a quantity under a radical.

2.Compute .3.Rewrite the integral in terms of the variable

u.4.Find the resulting integral in terms of u.5.Replace u by to obtain an

antiderivative in terms of x.

u g x

du g x dx

g x

Theorem: Change of Variables for Definite Integrals

g b

g a

f g x g x f u du

THE GENERAL POWER RULE FOR INTEGRATION

• If u is a function of x and n is not equal to -1, then

 

 

1

1

nn uu du C

n

Even Functions

2

-2

h y = 1

g y = -1

f x = -x2+3

0

2a a

a

f x dx f x dx

Odd Functions

0a

a

f x dx

2

-2

q y = 1

g y = -1

f x = 2x