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Published byOswald Edward Ramsey Modified over 5 years ago

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Drill: Find dy/dx y = x 3 sin 2x y = e 2x ln (3x + 1) y = tan -1 2x Product rule: x 3 (2cos 2x) + 3x 2 sin (2x) 2x 3 cos 2x + 3x 2 sin (2x) Product Rule e 2x (3/(3x +1)) + 2e 2x ln (3x + 1) 3e 2x /(3x +1) + 2e 2x ln (3x + 1)

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Antidifferentiation by Parts Lesson 6.3

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Objectives Students will be able to: – use integration by parts to evaluate indefinite and definite integrals. – use rapid repeated integration or tabular method to evaluate indefinite integrals.

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Integration by Parts Formula A way to integrate a product is to write it in the form If u and v are differentiable function of x, then

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Example 1 Using Integration by Parts Evaluate

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Example 1 Using Integration by Parts Evaluate

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Example 1 Using Integration by Parts Evaluate

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Example 2 Repeated Use of Integration by Parts Evaluate

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Example 2 Repeated Use of Integration by Parts Evaluate

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Example 3 Solving an Initial Value Problem Solve the differential equation dy/dx = xlnx subject to the initial condition y = -1 when x = 1 It is typically better to let u = lnx

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Drill Solve the differential equation: dy/dx = x 2 e 4x (This means you will need to find the anti-derivative of dy/dx = x 2 e 4x )

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Example 4 Solving for the unknown integral

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Rapid Repeated Integration by Parts AKA: The Tabular Method Choose parts for u and dv. Differentiate the u’s until you have 0. Integrate the dv’s the same number of times. Multiply down diagonals. Alternate signs along the diagonals.

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Example 5Rapid Repeated Integration by Parts Evaluate u and its derivativesdv and its integrals

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Example 5Rapid Repeated Integration by Parts Evaluate

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Example 5Rapid Repeated Integration by Parts Evaluate u and its derivativesdv and its integrals

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Example 5Antidifferentiating ln x

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Example 6Antidifferentiating sin -1 x

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Homework Page 346/7: Day #1: 1-15 odd Page 347: 17-24

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