Drill find dy dx
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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). Antidifferentiation by Parts. Lesson 6.3.

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Drill: Find dy / dx

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Drill find dy dx

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

Antidifferentiation by Parts

Lesson 6.3


Objectives

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

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

Example 1 Using Integration by Parts

Evaluate


Example 1 using integration by parts1

Example 1 Using Integration by Parts

Evaluate


Example 1 using integration by parts2

Example 1 Using Integration by Parts

Evaluate


Example 2 repeated use of integration by parts

Example 2 Repeated Use ofIntegration by Parts

Evaluate


Example 2 repeated use of integration by parts1

Example 2 Repeated Use ofIntegration by Parts

Evaluate


Example 3 solving an initial value problem

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

Drill

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


Example 4 solving for the unknown integral

Example 4Solving for the unknown integral


Rapid repeated integration by parts aka the tabular method

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 5 rapid repeated integration by parts

Example 5Rapid Repeated Integration by Parts

Evaluate


Example 5 rapid repeated integration by parts1

Example 5Rapid Repeated Integration by Parts

Evaluate


Example 5 rapid repeated integration by parts2

Example 5Rapid Repeated Integration by Parts

Evaluate


Example 5 antidifferentiating ln x

Example 5Antidifferentiatingln x


Example 6 antidifferentiating sin 1 x

Example 6Antidifferentiatingsin-1 x


Homework

Homework

  • Page 346/7: Day #1: 1-15 odd

  • Page 347: 17-24


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