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INTEGRATION BY SUBSTITUTION Section 4.5
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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
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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. 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.
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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
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PATTERN RECOGNITION Pattern Recognition applies the preceding theorem directly –We need to recognize and
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Which expression represents in the integral shown? A. B. C.
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Which expression represents in the integral shown? A. B. C.
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Guidelines for Making a Change of Variables 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. Compute. Rewrite the integral in terms of the variable u. Find the resulting integral in terms of u. Replace u by to obtain an antiderivative in terms of x.
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Theorem: Change of Variables for Definite Integrals
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Find the area under the curve bounded by the graph of,, and the x-axis and the y-axis. 9/4 0.0
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THE GENERAL POWER RULE FOR INTEGRATION If the function has a continuous derivative on the closed interval and f is continuous on the range of u, then
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Even Functions
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Odd Functions
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