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Integration by Parts: Learn the Formula and Examples for Quick Solutions

Understand Integration by Parts with the product rule, tackling Logs, Inverse trig, Polynomial, Exponential, and Trig functions. Explore LIPET method with examples and a shortcut using Tabular Integration for efficient solutions.

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Integration by Parts: Learn the Formula and Examples for Quick Solutions

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  1. Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington 6.3 Integration By Parts Badlands, South Dakota

  2. 6.3 Integration By Parts Start with the product rule: This is the Integration by Parts formula.

  3. 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

  4. Example 1: LIPET polynomial factor

  5. Example 4: LIPET logarithmic factor

  6. Example 5: LIPET This is still a product, so we need to use integration by parts again.

  7. Example 6: LIPET This is the expression we started with!

  8. Example 6: LIPET

  9. This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever. Example 6:

  10. A Shortcut: Tabular Integration Tabular integration works for integrals of the form: where: Differentiates to zero in several steps. Integrates repeatedly.

  11. Compare this with the same problem done the other way:

  12. Example 5: LIPET This is easier and quicker to do with tabular integration!

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