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今天小编为大家带来的是好学高数（十）：定积分。

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Dear you, this is LearningYard.

Today's little editor brings you the idea of learning advanced numbers (10): definite integral.

一、定积分的概念与性质

1、定义

可以看成求区间与函数图像围成的面积，但有正负之分。

2、定积分相当于一个数

一、 The concept and properties of definite integral 1. The definition can be seen as the area enclosed by interval and function image, but it can be positive or negative.

2. Definite integral is equivalent to a number

二、微积分的基本公式

1、牛顿-莱布尼茨公式

二、 Basic Formula of Calculus

1. Newton Leibniz Formula

三、定积分的换元法和分部积分法

1、与不定积分类似，不同的是需要运用牛顿-莱布尼茨公式求出具体的数（这里也可以说明定积分是一个数）

2、计算方法

①计算时观察函数奇偶性，偶倍奇零。

②观察函数的周期性

三、 The substitution method and partial integration method of definite integral

1. Similar to indefinite integral, the difference is that the Newton Leibniz formula needs to be used to find the specific number (here it can also be explained that the definite integral is a number)

2. Calculation method

① Observe the parity of the function when calculating, even times odd zero.

② Observe the periodicity of a function

四、反常积分

1、无穷限的反常积分

①定义

②注意事项：当区间为负无穷到正无穷，取c为任意常数，将原式分为两个极限进行相加，这时只要有一个极限不存在，原式就发散。出现无穷减无穷的结果，并非不定型。

③易错点：对于反常积分，只有在收敛的条件下才能使用偶倍奇零。

所以对反常积分，只有在收敛条件下才能使用“偶倍奇零”的性质。

2、无界函数的反常积分

①定义

②可借助牛顿-莱布尼茨公式进行计算

四. Abnormal integral

1. Abnormal integral of infinite limit

① Definition

② Precautions: when the interval is from negative infinity to positive infinity, c is taken as an arbitrary constant, and the original formula is divided into two limits for addition. At this time, as long as one limit does not exist, the original formula will diverge. The result of infinity minus infinity is not indefinite.

③ Error prone point: For abnormal integrals, even odd zeros can only be used under the condition of convergence. Therefore, for abnormal integrals, the property of "even times odd zero" can only be used under the condition of convergence.

2. Abnormal integral of unbounded function

① definition

② can be calculated with Newton Leibniz formula

END

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翻译：百度翻译

参考：《高等数学 第七版》同济大学数学系编

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文案&排版：易春秀

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