THE INDEFINITE INTEGRAL Chapter 5 THE INDEFINITE INTEGRAL

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

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

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

Question? Solution

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.

3. Examples of integration by parts

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

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

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.

Example 3 Find Solution: Let

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

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:

Example 6 Find Solution: circulation

Example 7 Find Solution:

Let

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

Example 9 Solution:

Example 10 Solution:

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