Calculus 6.3

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Re-write using a substitution. Do not integrate.
??????
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7.1 Integration By Parts
Start with the product rule
This is the Integration by Parts formula.
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Or for a definite integral
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
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When do you use Integration by Parts (IBP)?

• Usually, the function is the product of 2
• different types of functions. Polynomial,
• log, exponential, trig, etc.

• U-substitution does not produce a known
• pattern to integrate.

2 different types of functions, but
2 different types of functions, and no u-sub
works.
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Example 1
LIPET
polynomial factor
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Example
LIPET
logarithmic factor
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Example 4
LIPET
This is still a product, so we need to use
integration by parts again.
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Example 5
LIPET
This is the expression we started with!
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Example 6
LIPET
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Example 6
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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 5
LIPET
This is easier and quicker to do with tabular
integration!
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p
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Doesnt follow any pattern for integration that
we know.
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What if
So instead
Then
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Homework Page 492 5, 11, 19, 27, 31, 35, 39,
57, 61, 64b