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Sec. 4.5: Integration by Substitution

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T HEOREM 4.12 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

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Sec. 4.5: Integration by Substitution

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Rewrite the integral:

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Sec. 4.5: Integration by Substitution When solving a differential equation of the form it’s convenient to write it of the form

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Sec. 4.5: Integration by Substitution Find the differential dy.

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Sec. 4.5: Integration by Substitution u = 7x du = 7 dx

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Sec. 4.5: Integration by Substitution u = x 2 – 4 du = 2x dx 1/2 du = x dx

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Sec. 4.5: Integration by Substitution u = sin 3x du = (cos 3x) 3 dx 1/3 du = cos 3x dx

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Sec. 4.5: Integration by Substitution u = 5x + 3 du = 5 dx 1/5 du = dx

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Sec. 4.5: Integration by Substitution Rearranging u u = x + 3 du = dx u – 3 = x

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Sec. 4.5: Integration by Substitution u = 2x – 1 du = 2 dx u + 1 = 2x 1/2 du = dx (u + 1) / 2 = x

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1 5.b – The Substitution Rule. 2 Example – Optional for Pattern Learners 1. Evaluate 3. Evaluate Use WolframAlpha.com to evaluate the following. 2. Evaluate.

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