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

Objectives Find the antiderivative using integration by parts. Use a tabular method to perform integration by parts.

Graphing Calculator Activity: Graph and find the area bounded by and Window: xmin=-1.88 ymin=-1.2 xmax=1.88 ymax=1.7 A valentine for your sweetie "pi". fnInt(y2-y1,x,-1,1) Shade(y1,y2) {Draw 7}

How do you integrate

7.2 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: LIATE Logs, Inverse trig, Algebraic, Trig ,Exponential

LIATE

LIATE

Can't integrate arcsin!

Example 1: LIATE polynomial factor

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

A Shortcut: Tabular Integration (Tic-Tac-Toe Method) 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: This is easier and quicker to do with tabular integration!

Homework Handout #1-15 odd 21, 29, 31, 35 p