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Integration by Parts Sec. 6.2

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

u differentiates to zero (usually). 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

Example 1: LIPET polynomial factor

Example: LIPET logarithmic factor

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

Example 7:

Example 8:

Example 5: LIPET This is the expression we started with!

Example 6: LIPET

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

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

Compare this with the same problem done the other way:

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

p

Homework P. 346-347 #1-35 odd Quiz on Integrating tomorrow (definite and indefinite, u-substitution, initial value problems)