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# Drill: Find dy / dx - PowerPoint PPT Presentation

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.

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Drill: Find dy/dx

• y = x3 sin 2x

• y = e2xln (3x + 1)

• y = tan-1 2x

• Product rule:

• x3 (2cos 2x) + 3x2 sin (2x)

• 2x3cos 2x + 3x2 sin (2x)

• Product Rule

• e2x (3/(3x +1)) + 2e2xln (3x + 1)

• 3e2x/(3x +1) + 2e2xln (3x + 1)

### Antidifferentiation by Parts

Lesson 6.3

• 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.

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

Solve the differential equation: dy/dx= x2e4x (This means you will need to find the anti-derivative of dy/dx = x2e4x)

Example 4Solving for the unknown integral

Rapid Repeated Integration by PartsAKA: 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

Example 5Rapid Repeated Integration by Parts

Evaluate

Example 5Rapid Repeated Integration by Parts

Evaluate

Example 5Antidifferentiatingln x

Example 6Antidifferentiatingsin-1 x

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