# 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.

## Presentation on theme: "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."— Presentation transcript:

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)

Antidifferentiation by Parts Lesson 6.3

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.

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

Example 1 Using Integration by Parts Evaluate

Example 1 Using Integration by Parts Evaluate

Example 1 Using Integration by Parts Evaluate

Example 2 Repeated Use of Integration by Parts Evaluate

Example 2 Repeated Use of Integration by Parts Evaluate

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

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 )

Example 4 Solving for the unknown integral

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.

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

Example 5Rapid Repeated Integration by Parts Evaluate

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

Example 5Antidifferentiating ln x

Example 6Antidifferentiating sin -1 x

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

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