1. 8.2
Integration By Parts

2. Start with the product rule:

3. Integration by Parts 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 still a product, so we need to use integration by parts again.

7. Example

8. A Shortcut: Tabular Integration
Tabular integration works for integrals of the form:
where:
Differentiates to 0 in several steps.
Integrates repeatedly.
Example:

9. Example This is the expression we started with!
This is called “solving
for the unknown integral.”
It works when both factors integrate and differentiate forever.

10. How to choose u and dv
Try to choose u so that du (its derivative) becomes easier to integrate than u.
If ln is present, then u must be ln.
Oftentimes, let u be the powers of x.
Also, choose dv so that it is easy to integrate dv.
If ex is present, let dv = ex dx
Oftentimes, let dv be the sin or cos.
After integrating by parts, you should wind up with the integral that is “easier” to integrate.

11. Examples