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Mastering Integration by Parts: A Comprehensive Guide

Learn the powerful Integration by Parts method starting with the product rule. Explore examples with various types of functions such as logarithmic, polynomial, exponential, and trigonometric. Discover a shortcut with the Tabular Method for quick and efficient integration.

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Mastering Integration by Parts: A Comprehensive Guide

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  1. 6.3 Integration By Parts

  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: LIPET logarithmic factor

  6. Example 4: LIPET This is still a product, so we need to use integration by parts again.

  7. Example 5: 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. Tabular Method

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

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

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

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