6.3 Integration By Parts

Badlands, South Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993

6.3 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

Example 1:

polynomial factor

LIPET

Example:

logarithmic factor

LIPET

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

Example 4: LIPET

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