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8.2 Integration By Parts

8.2 Integration By Parts. Start with the product rule:. Integration by Parts. 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.

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8.2 Integration By Parts

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  1. 8.2 Integration By Parts

  2. Start with the product rule:

  3. Integration by Parts 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 Logs, Inverse trig, Polynomial, Exponential, Trig

  4. Example LIPET polynomial factor

  5. Example LIPET logarithmic factor

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

  7. Example

  8. A Shortcut: Tabular Integration Tabular integration works for integrals of the form: where: Differentiates to 0 in several steps. Integrates repeatedly. Example:

  9. This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever. Example This is the expression we started with!

  10. How to choose u and dv • Try to choose u so that du (its derivative) becomes easier to integrate than u. • If ln is present, then u must be ln. • Oftentimes, let u be the powers of x. • Also, choose dv so that it is easy to integrate dv. • Ifexis present, let dv = exdx • Oftentimes, let dv be the sin or cos. • After integrating by parts, you should wind up with the integral that is “easier” to integrate.

  11. Examples

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