1. 6.3 Integration by Parts & Tabular Integration

2. Problem: Integrate Antiderivative is not obvious
U-substitution does not work
We must have another method to at least try and find the antiderivative!!!

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

4. 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

5. Example 1:
LIPET
polynomial factor

6. Example 2:
LIPET
logarithmic factor

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

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

9. Example 5:
LIPET

10. Example 5 (cont.):
This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.

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

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

13. Example 6:
LIPET
This is easier and quicker to do with tabular integration!