Improper Integral

Core Sentence / 核心句

An improper integral is a limit of proper definite integrals over an unbounded interval or near an unbounded integrand.

中文理解：反常积分不是把无穷当作普通上下限，而是用一列正常定积分的极限来定义无限区间或奇点附近的累计量。

Definitions / 定义

$${\int }_a^{\infty }f(x)dx:=\underset{b\to \infty }{\lim }{\int }_a^{b}f(x)dx.$$

For an interior singularity $c$,

$${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f,$$

with each piece defined by its own one-sided limit.

Convergence Rule / 收敛规则

The improper integral converges only when every defining limit exists as a finite real number. If any required piece diverges, the whole integral diverges.

Benchmark / 基准

$${\int }_1^{\infty }\frac{1}{x^p}dx\text{ converges iff }p>1.$$

Watch Out / 易错点

• Never substitute $\infty$ as if it were a number.
• Split $(-\infty ,\infty )$ and test both tails separately.
• Split at every vertical asymptote inside the interval.
• Divergent pieces cannot be canceled in the ordinary improper-integral definition.
• Antiderivative existence does not guarantee convergence; the limit decides.

Source / 出现位置

• 第9课 · 无限区间定义
• 第9课 · 分别检查两端
• 第9课 · 内部奇点

Connected Notes / 连接

• Limits
• Definite Integral
• Integrals
• p-Series
• Integral Test
• 微积分BC第9次正课-积分方法与反常积分
• 微积分BC第13次正课-无穷级数判敛