1. THE INDEFINITE INTEGRAL
Chapter 5
THE INDEFINITE INTEGRAL

2. New Words Reduction formula 递推公式 Recursion formula 递推公式
Integration by parts 分部积分法
Integrate 积分 radical 根号
Perfect square 完全平方
Hyperbolic substitution 双曲替换

3. 5.3 Integration by parts
This method is called integration by parts.

4. 1. Integration by parts
We will first illustrate the method by an example. Right after the example we will explain why (1) is valid

5. Question?
Solution

6. By integration by parts,
2. The proof of integration by parts
Just as the chain rule is the basis for integration by substitution, the formula for the derivative of a product is the basis for integration by parts.

7. 3. Examples of integration by parts

8. Example 1 Find
Solution (1)
Let
Obviously, if we choice impropriety is difficult to integral
Solution (2)
Let

9. Remark:
(2) du should not be messier than u.

10. Example 2 Find
Solution
（Using again the integration by parts）
Summary
If the integrand is the product of power functions and sine or cosine functions, power functions and exponential functions, we can let u as power function.

11. Example 3 Find
Solution:
Let

12. Example 4 Find
Solution:
Summary
If the integrand is the product of power function and logarithm function, or power function and inverse trigonometric function, we can let u as logarithm function or inverse trigonometric function

13. In the following examples one integration by parts appears at first to be useless, but two in succession find the integral.
Example 5 Find
Solution:

14. Example 6 Find
Solution:
circulation

15. Example 7 Find
Solution:

16. Let

17. Reduction formulas
We can get some reduction formulas or recursion formulas by an integration by parts.
Example 8
Solution:

19. Example 9
Solution:

21. Example 10
Solution:

22. See you next time