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

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: LIPET polynomial factor

Example: LIPET logarithmic factor

Example: LIPET This is the expression we started with!

Example: LIPET

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

A Shortcut: Tabular Integration Compare this with the same problem done the other way:

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

Nearly all integration by parts problems can be done using the tabular method, even if one term doesn’t differentiate down to zero: Example:

Notice this is something Example: Notice this is something you can integrate!

 Proof that “one actually equals zero” So… Therefore: Evaluate using integration by parts: So… Therefore: 