1. 6.3 Integration By Parts Start with the product rule:
This is the Integration by Parts formula.

2. 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

3. Example 1:
LIPET
polynomial factor

4. Example:
LIPET
logarithmic factor

5. Example 4:
LIPET
This is still a product, so we need to use integration by parts again.

6. Example 5:
LIPET
This is the expression we started with!

7. Example 6:
LIPET

8. Example 6:
This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.

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

10. Compare this with the same problem done the other way:

11. Example 5:
LIPET
This is easier and quicker to do with tabular integration!

12. p