7.1 Integration By Parts

Start with the product rule:

Integration by Parts 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 still a product, so we need to use integration by parts again.

Example

A Shortcut: Tabular Integration Tabular integration works for integrals of the form: where: Differentiates to 0 in several steps. Integrates repeatedly. Example:

Example This is the expression we started with! This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever.

How to choose u and dv Try to choose u so that du (its derivative) becomes easier to integrate than u. If ln is present, then u must be ln. Oftentimes, let u be the powers of x. Also, choose dv so that it is easy to integrate dv. If ex is present, let dv = ex dx Oftentimes, let dv be the sin or cos. After integrating by parts, you should wind up with the integral that is “easier” to integrate.

Examples