1. 8.2 Integration By Parts Badlands, South Dakota
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

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

3. 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

4. Example:
LIPET
polynomial factor

5. Example:
LIPET
logarithmic factor

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

7. Example:
LIPET

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

9. A Shortcut: Tabular Integration
Compare this with the same problem done the other way:

10. LIPET
This is easier and quicker to do with tabular integration!

11. Nearly all integration by parts problems can be done using
the tabular method, even if one term doesn’t differentiate down
to zero:
Example:

13. Notice this is something
Example:
Notice this is something
you can integrate!

14.  Proof that “one actually equals zero” So… Therefore: Evaluate
using integration by parts:
So…
Therefore: 