INTEGRATION BY SUBSTITUTION Section 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 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

PATTERN RECOGNITION We need to recognize and

Which expression represents in the integral shown? A. B. C.

Which expression represents in the integral shown? A. B. C.

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.

Theorem: Change of Variables for Definite Integrals

THE GENERAL POWER RULE FOR INTEGRATION If u is a function of x and n is not equal to -1, then

Even Functions

Odd Functions