Improper Integrals: Infinite Limits and Unbounded Integrands | 反常积分：无穷限与无界被积函数

📚 Improper Integrals: Infinite Limits and Unbounded Integrands | 反常积分：无穷限与无界被积函数

In AQA A-Level Mathematics, an improper integral is a definite integral that cannot be evaluated directly because one or both limits are infinite, or because the integrand becomes unbounded within the interval of integration. This topic brings together limits, antiderivatives and the Fundamental Theorem of Calculus. The key idea is always the same: rewrite the improper integral as a limit of a proper integral, then decide whether the limit exists as a finite number.

在 AQA A-Level 数学中，反常积分是指不能直接求值的定积分，因为一个或两个积分限为无穷大，或者被积函数在积分区间内无界。这一主题将极限、原函数和微积分基本定理结合在一起。核心思想始终相同：把反常积分改写为正常积分的极限，然后判断该极限是否存在且为有限值。

1. What Is an Improper Integral? | 什么是反常积分

A proper definite integral has finite limits of integration and a bounded integrand on the interval. An improper integral arises when at least one of these conditions fails. The two main types are infinite limits of integration and infinite discontinuities of the integrand. In both cases, the Fundamental Theorem of Calculus cannot be applied directly.

正常定积分具有有限的积分限，并且被积函数在积分区间上有界。只要其中一个条件不满足，就会产生反常积分。两种主要类型是无穷积分限和被积函数存在无穷间断点。在这两种情况下，都不能直接使用微积分基本定理。

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