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Drill: Find dy / dx

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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  1. 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)

  2. Antidifferentiation by Parts Lesson 6.3

  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.

  4. 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

  5. Example 1 Using Integration by Parts Evaluate

  6. Example 1 Using Integration by Parts Evaluate

  7. Example 1 Using Integration by Parts Evaluate

  8. Example 2 Repeated Use of Integration by Parts Evaluate

  9. Example 2 Repeated Use of Integration by Parts Evaluate

  10. 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

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

  12. Example 4Solving for the unknown integral

  13. 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.

  14. Example 5Rapid Repeated Integration by Parts Evaluate

  15. Example 5Rapid Repeated Integration by Parts Evaluate

  16. Example 5Rapid Repeated Integration by Parts Evaluate

  17. Example 5Antidifferentiatingln x

  18. Example 6Antidifferentiatingsin-1 x

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

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