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8.2 Integration By Parts

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**Start with the product rule:**

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**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

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Example LIPET polynomial factor

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Example LIPET logarithmic factor

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

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Example

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**A Shortcut: Tabular Integration**

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

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**Example This is the expression we started with!**

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

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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. If ex is present, let dv = ex dx Oftentimes, let dv be the sin or cos. After integrating by parts, you should wind up with the integral that is “easier” to integrate.

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Examples

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