# 8.2 Integration By Parts Badlands, South Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

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8.2 Integration By Parts Badlands, South Dakota Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

Integrate the following:

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

The Integration by Parts formula is a “product rule” for integration. u differentiates to zero (usually). dv is easy to integrate. Choose u in this order: LIPET Logs, Inverse trig, Polynomial, Exponential, Trig Or LIPTE

Integration by Parts !

Example 1: polynomial factor LIPET

Example 2: logarithmic factor LIPET

This is still a product, so we need to use integration by parts again. Example 3: LIPET

Example 4: LIPET This is the expression we started with!

Example 4 (con’t):

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

Integration by Parts

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

Compare this with the same problem done the other way:

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

Find You Try:

Solution: Begin as usual by letting u = x 2 and dv = v' dx = sin 4x dx. Next, create a table consisting of three columns, as shown.

Homework: Day 1: pg. 531, 11-55 EOO, 59-69 odd. Day 2: MMM BC pgs. 106-107

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