1. TECHNIQUES OF INTEGRATION
In this chapter we develop techniques for using these basic integration formulas to obtain indefinite integrals of more complicated functions.

2. Find Find INTEGRATION BY PARTS formula for integration by parts.
Example
Example
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Find

3. Find INTEGRATION BY PARTS formula for integration by parts.
REMARK1: aim in using integration by parts is to obtain a simpler integral than the one we started with.
REMARK2: How to choose u and dv to obtain simpler integral
Example
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4. INTEGRATION BY PARTS
Example
REMARK2: in some integral, we may need to apply integration by parts many times.
Find

5. INTEGRATION BY PARTS
Example
REMARK2: in some integral, we may need to apply integration by parts many times.
Find
Example
Find
Example
Find

6. INTEGRATION BY PARTS
formula for integration by parts.
Example
Find
REMARK3: sometimes a repeated application of integration by parts leads back to an integral similar to our original one. If so, this expression can be combined with original integral.

7. INTEGRATION BY PARTS
Exam2
Term082
Exam2
Term102

8. Exam2
Term092

10. INTEGRATION BY PARTS
Observe:
Reduction Formula
REMARK3: sometimes The reduction formula is useful because by using it repeatedly we could eventually express our integral.

11. INTEGRATION BY PARTS
Reduction Formula
Example
Example

12. Reduction Formula Reduction Formula INTEGRATION BY PARTS Example

13. Reduction Formula

14. Most of the time ln(x) is easier by parts Repeated Applications
INTEGRATION BY PARTS
Most of the time ln(x) is easier by parts
Reduction Formula
Integration by Parts
Repeated Applications
Term 0 (diff)
tabular integration
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