8.2 Integration by parts

Formula for Integration by parts The idea is to use the above formula to simplify an integration task. One wants to find a representation for the function to be integrated in the form udv so that the function vdu is easier to integrate than the original function. The rule is proved using the Product Rule for differentiation.

Deriving the Formula Start with the product rule: This is the Integration by Parts formula.

Choosing u and v dv is easy to integrate. u differentiates to zero (usually). Choose u in this order: LIPET Logs, Inverse trig, Polynomial, Exponential, Trig

Example 1: LIPET polynomial factor

Example 2: LIPET logarithmic factor

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

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

Example 4(cont.): LIPET This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever.

Integration by Parts for Definite Integrals Formula Integration by Parts Formula and the Fundamental Theorem of Calculus imply the above Integration by Parts Formula for Definite Integrals. Here we must assume that the functions u and v and their derivatives are all continuous. Example

Integration by Parts for Definite Integrals Example (cont’d) By the computations on the previous slide we now have Combining these results we get the answer