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Integration By Parts

Integration By Parts. 1. 1. Suppose we want to integrate this function. Up until now we have no way of doing this. x , and sin(x) seem totally unrelated. If u(x) is a function and v(x) is another function we seem to have

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Integration By Parts

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  1. Integration By Parts (c) 2014 joan s kessler distancemath.com 1 1

  2. Suppose we want to integrate this function. Up until now we have no way of doing this. x, and sin(x) seem totally unrelated. If u(x) is a function and v(x) is another function we seem to have It almost seems like the reverse of the product rule. Let’s explore the product rule. (c) 2014 joan s kessler distancemath.com 2

  3. Integration By Parts Start with the product rule: This is the Integration by Parts formula. (c) 2014 joan s kessler distancemath.com 3

  4. Logs, Inverse trig, Polynomial, Exponential, Trig dv is easy to integrate. u differentiates to zero (usually). The Integration by Parts formula is a “product rule” for integration. Choose u in this order: LIPET (c) 2014 joan s kessler distancemath.com 4

  5. Example 1: LIPET polynomial factor (c) 2014 joan s kessler distancemath.com 5

  6. Example 2: LIPET logarithmic factor (c) 2014 joan s kessler distancemath.com 6

  7. Example 3: LIPET This is still a product, so we need to use integration by partsagain. (c) 2014 joan s kessler distancemath.com 7

  8. Example 4: LIPET This is the expression we started with! (c) 2014 joan s kessler distancemath.com 8

  9. Example 5: LIPET (c) 2014 joan s kessler distancemath.com 9

  10. This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever. Example 5 : (c) 2014 joan s kessler distancemath.com 10

  11. More integration by Parts Ex 6. Let u = arcsin3x dv = dx v = x -18 -118 (c) 2014 joan s kessler distancemath.com 11

  12. More integration by Parts Ex. 7 Let u = x2 du = 2x dx Form (c) 2014 joan s kessler distancemath.com 12

  13. A Shortcut: Tabular Integration Tabular integration works for integrals of the form: where: Differentiates to zero in several steps. Integrates repeatedly. (c) 2014 joan s kessler distancemath.com 13

  14. Alternate signs Compare this with the same problem done the other way: (c) 2014 joan s kessler distancemath.com 14

  15. Same Example : LIPET This is easier and quicker to do with tabular integration! (c) 2014 joan s kessler distancemath.com 15

  16. Factor the answer if possible (c) 2014 joan s kessler distancemath.com 16

  17. (c) 2014 joan s kessler distancemath.com 17

  18. (c) 2014 joan s kessler distancemath.com

  19. (c) 2014 joan s kessler distancemath.com 19

  20. Factor the answer if possible (c) 2014 joan s kessler distancemath.com 20

  21. Try (c) 2014 joan s kessler distancemath.com 21

  22. +- + - + (c) 2014 joan s kessler distancemath.com 22

  23. +- + - + (c) 2014 joan s kessler distancemath.com 23

  24. +- + - (c) 2014 joan s kessler distancemath.com

  25. Homework Assignment

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