6.3 Integration By Parts Badlands, South Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993
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6.3 Integration By Parts
Badlands, South Dakota Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993
6.3 Integration By Parts
Start with the product rule:
This is the Integration by Parts formula.
The Integration by Parts formula is a “product rule” for integration.
u differentiates to zero (usually).
dv is easy to integrate.
Choose u in this order: LIPET
Logs, Inverse trig, Polynomial, Exponential, Trig
Example 1:
polynomial factor
LIPET
Example:
logarithmic factor
LIPET
This is still a product, so we need to use integration by parts again.
Example 4: LIPET
Example 5: LIPET
This is the expression we started with!
Example 6: LIPET
Example 6: This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.
A Shortcut: Tabular Integration
Tabular integration works for integrals of the form:
where: Differentiates to zero in several steps.
Integrates repeatedly.
Compare this with the same problem done the other way:
Example 5: LIPET
This is easier and quicker to do with tabular integration!
p
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