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6.3 Integration by Parts & Tabular Integration
Problem: Integrate Antiderivative is not obvious
U-substitution does not work We must have another method to at least try and find the antiderivative!!!
By Parts formula: Start with the product rule:
This is the Integration by Parts formula.
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
Example 1: LIPET polynomial factor
Example 2: LIPET logarithmic factor
Example 3: LIPET This is still a product, so we need to use integration by parts again.
Example 4: LIPET This is the expression we started with!
Example 5: LIPET
Example 5 (cont.): 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 6: LIPET This is easier and quicker to do with tabular integration!
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