1. Integration By Parts
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2. Suppose we want to integrate this function.
Up until now we have no way of doing this.
x, and sin(x) seem totally unrelated.
If u(x) is a function and v(x) is another function we seem to have
It almost seems like the reverse of the product rule. Let’s explore the product rule.
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3. Integration By Parts Start with the product rule:
This is the Integration by Parts formula.
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4. 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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5. Example 1: LIPET polynomial factor 5
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6. Example 2: LIPET logarithmic factor 6
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7. This is still a product, so we need to use integration by parts again.
Example 3:
LIPET
This is still a product, so we need to use integration by parts again.
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8. This is the expression we started with!
Example 4:
LIPET
This is the expression we started with!
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9. Example 5:
LIPET
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10. Example 5 : This is called “solving for the unknown integral.”
It works when both factors integrate and differentiate forever.
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11. More integration by Parts Ex 6.
Let u = arcsin3x dv = dx
v = x
-18
-1 18
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12. More integration by Parts Ex. 7
Let u = x du = 2x dx
Form
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13. 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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14. Compare this with the same problem done the other way:
Alternate signs
Compare this with the same problem done the other way:
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15. This is easier and quicker to do with tabular integration!
Same Example :
LIPET
This is easier and quicker to do with tabular integration!
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16. Factor the answer if possible
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20. Factor the answer if possible
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21. Try
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22. + - + -
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23. + - + -
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24. + - + -
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25. Homework Assignment