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6.3 Integration by Parts & Tabular Integration

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Problem: Integrate Antiderivative is not obvious

U-substitution does not work We must have another method to at least try and find the antiderivative!!!

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By Parts formula: Start with the product rule:

This is the Integration by Parts formula.

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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

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Example 1: LIPET polynomial factor

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Example 2: LIPET logarithmic factor

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Example 3: LIPET This is still a product, so we need to use integration by parts again.

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Example 4: LIPET This is the expression we started with!

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Example 5: LIPET

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Example 5 (cont.): This is called “solving for the unknown integral.” It works when both factors integrate and differentiate forever.

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A Shortcut: Tabular Integration

Tabular integration works for integrals of the form: where: Differentiates to zero in several steps. Integrates repeatedly.

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Compare this with the same problem done the other way:

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

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