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## PowerPoint Slideshow about ' Drill: Find dy / dx' - gianna

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### Antidifferentiation by Parts

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)

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= 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

Homework

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

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