1. Re-write using a substitution. Do not integrate.
??????

2. 7.1 Integration By Parts Start with the product rule:
This is the Integration by Parts formula.

3. u differentiates to zero (usually).
Or for a definite integral
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

4. When do you use Integration by Parts (IBP)?
Usually, the function is the product of 2
different types of functions. Polynomial,
log, exponential, trig, etc.
U-substitution does not produce a known
pattern to integrate.
2 different types of functions, but…
2 different types of functions, and no
u-sub works.

5. Example 1:
LIPET
polynomial factor

6. Example:
LIPET
logarithmic factor

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

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

9. Example 6:
LIPET

10. Example 6:
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 5:
LIPET
This is easier and quicker to do with tabular integration!

14. p

15. Doesn’t follow any pattern for integration that we know.

16. What if…
So instead:
Then…

18. Homework: Page 492
5, 11, 19, 27, 31, 35, 39, 57, 61, 64b