反常积分

发表: 5/23/2026 更新: 6/1/2026 字数: 0 字 时长: 0 分钟

在定积分的定义一节我们提到了黎曼积分可积的一个充分条件:

$\mathrm{若}f(x)\mathrm{在}[a,b]\mathrm{上}\mathrm{有}\mathrm{界},\mathrm{且}\mathrm{存}\mathrm{在}\mathrm{有}\mathrm{限}\mathrm{个}\mathrm{第}\mathrm{一}\mathrm{类}\mathrm{间}\mathrm{断}\mathrm{点},\mathrm{则}f(x)\mathrm{必}\mathrm{可}\mathrm{积}$

现在我们定义反常积分:
① $f(x)$ 积分区间无限
② $f(x)$ 在 $[a,b]$ 上有瑕点

满足任意一点即称为反常积分.

积分区间无限

1. $f(x)\in c[a,+\mathrm{\infty })$

def - ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x=F(b)-F(a)}$
Notes:

① ${\displaystyle F(b)-F(a)\ne {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x}$
② ${\displaystyle \underset{b\to +\mathrm{\infty }}{lim}[F(b)-F(a)]={\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x}$

If $\underset{b\to +\mathrm{\infty }}{lim}[F(b)-F(a)]=A.$ then 称 ${\displaystyle {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x}$ 收敛于A，记 ${\displaystyle {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x=A;}$
If $\underset{b\to +\mathrm{\infty }}{lim}[F(b)-F(a)]$ 不存在. then 称 ${\displaystyle {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x}$ 发散.

2. $f(x)\in c(-\mathrm{\infty },a]$

def - ${\displaystyle {\int }_b^{a}f(x)\mathrm{d}x=F(a)-F(b)}$
Notes:

① ${\displaystyle F(a)-F(b)\ne {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x}$
② ${\displaystyle \underset{b\to -\mathrm{\infty }}{lim}[F(a)-F(b)]={\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x}$

If $\underset{b\to -\mathrm{\infty }}{lim}[F(a)-F(b)]=A.$ then 称 ${\displaystyle {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x}$ 收敛于A，记 ${\displaystyle {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x=A;}$
If $\underset{b\to -\mathrm{\infty }}{lim}[F(a)-F(b)]$ 不存在. then 称 ${\displaystyle {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x}$ 发散.

3. $f(x)\in c(-\mathrm{\infty },+\mathrm{\infty }]$

def - ${\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(x)\mathrm{d}x={\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x+{\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x}$

极限形式判别法

1. $f(x)\in c[a,+\mathrm{\infty })(a>0)$
   ① ${\displaystyle \mathrm{\exists }\alpha >1:\underset{x\to +\mathrm{\infty }}{lim}{x}^{\alpha }f(x)\mathrm{\exists }\Rightarrow {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛};}$
   ② ${\displaystyle \mathrm{\exists }\alpha \le 1:\underset{x\to +\mathrm{\infty }}{lim}{x}^{\alpha }f(x)=k(\ne 0)\mathrm{或}\mathrm{\infty }\Rightarrow {\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x\mathrm{发}\mathrm{散}.}$

2. $f(x)\in c(-\mathrm{\infty },a](a<0)$
   ① ${\displaystyle \mathrm{\exists }\alpha >1:\underset{x\to -\mathrm{\infty }}{lim}{x}^{\alpha }f(x)\mathrm{\exists }\Rightarrow {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛};}$
   ② ${\displaystyle \mathrm{\exists }\alpha \le 1:\underset{x\to -\mathrm{\infty }}{lim}{x}^{\alpha }f(x)=k(\ne 0)\mathrm{或}\mathrm{\infty }\Rightarrow {\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x\mathrm{发}\mathrm{散}.}$

3. $f(x)\in c(-\mathrm{\infty },+\mathrm{\infty }]$

Conc ${\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛}⟺{\int }_{-\mathrm{\infty }}^{a}f(x)\mathrm{d}x\mathrm{与}{\int }_a^{+\mathrm{\infty }}f(x)\mathrm{d}x\mathrm{都}\mathrm{收}\mathrm{敛}.}$

对于反常积分来说，敛散性是高于奇偶性的，只有确定敛散性后才能使用奇偶性.

例1. ${\displaystyle {\int }_0^{+\mathrm{\infty }}\frac{\sqrt{x}}{1+x^2}\mathrm{d}x?}$
$\begin{array}{l}{\displaystyle \because \underset{x\to +\mathrm{\infty }}{lim}{x}^{\frac{3}{2}}\cdot \frac{\sqrt{x}}{1+x^2}=\underset{x\to +\mathrm{\infty }}{lim}\frac{x^2}{1+x^2}=1\mathrm{且}\alpha =\frac{3}{2}>1}\\ {\displaystyle \therefore {\int }_0^{+\mathrm{\infty }}\frac{\sqrt{x}}{1+x^2}\mathrm{d}x\mathrm{收}\mathrm{敛}.}\end{array}$

例2. ${\displaystyle {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{x}{1+x^2}\mathrm{d}x?}$
$\begin{array}{l}{\displaystyle \because \underset{x\to +\mathrm{\infty }}{lim}{x}^1\cdot \frac{x}{1+x^2}=1(\ne 0)\mathrm{且}\alpha =1\le 1}\\ {\displaystyle \therefore {\int }_0^{+\mathrm{\infty }}\frac{x}{1+x^2}\mathrm{d}x\mathrm{发}\mathrm{散}}\\ {\displaystyle \therefore {\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{x}{1+x^2}\mathrm{d}x\mathrm{发}\mathrm{散}.}\end{array}$

⭐Gamma 函数

Note: $\mathrm{\Gamma }\mathrm{函}\mathrm{数}$(数一、三(概率统计))
def - ${\displaystyle \mathrm{\Gamma }(\alpha )\triangleq {\int }_0^{+\mathrm{\infty }}{x}^{\alpha -1}{e}^{-x}\mathrm{d}x(\alpha >0)}$
Nat -

① $\mathrm{\Gamma }(\alpha +1)=\alpha \mathrm{\Gamma }(\alpha );$
② $\mathrm{\Gamma }(n+1)=n!;$
③ $\mathrm{\Gamma }(\frac{1}{2})=\sqrt{\pi }$

积分区间有限

1. 瑕点

def -

若函数 $f(x)$ 在 $x=a$ 的任一领域内都无界，则称点 $a$ 为函数 $f(x)$ 的瑕点.
此时 $f(a\pm 0)$ 不存在.

当且仅当该点是无穷间断点或无界震荡间断点时为瑕点.

2. $f(x)\in c[a,b]$，且 $a$ 为唯一瑕点

def - ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}{\int }_{a+\epsilon }^{b}f(x)\mathrm{d}x=F(b)-F(a+\epsilon )}$
Notes:

① ${\displaystyle F(b)-F(a+\epsilon )\ne {\int }_a^{b}f(x)\mathrm{d}x}$
② ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b)-F(a+\epsilon )]={\int }_a^{b}f(x)\mathrm{d}x}$

If ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b)-F(a+\epsilon )]=A.}$ then 称 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x}$ 收敛于A，记 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x=A;}$
If ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b)-F(a+\epsilon )]}$ 不存在 then 称 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x}$ 发散.

3. $f(x)\in c[a,b]$，且 $b$ 为唯一瑕点

def - ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}{\int }_a^{b-\epsilon }f(x)\mathrm{d}x=F(b-\epsilon )-F(a)}$
Notes:

① ${\displaystyle F(b-\epsilon )-F(a)\ne {\int }_a^{b}f(x)\mathrm{d}x}$
② ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b-\epsilon )-F(a)]={\int }_a^{b}f(x)\mathrm{d}x}$

If ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b-\epsilon )-F(a)]=A.}$ then 称 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x}$ 收敛于A，记 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x=A;}$
If ${\displaystyle \underset{\epsilon \to 0^{+}}{lim}[F(b-\epsilon )-F(a)]}$ 不存在 then 称 ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x}$ 发散.

极限形式判别法

1. $f(x)\in c[a,b],a\mathrm{为}\mathrm{唯}\mathrm{一}\mathrm{瑕}\mathrm{点}(\mathrm{无}\mathrm{穷}\mathrm{间}\mathrm{断}\mathrm{点})$
   ① ${\displaystyle \mathrm{\exists }\alpha <1:\underset{x\to a+}{lim}(x-a{)}^{\alpha }f(x)\mathrm{\exists }\Rightarrow {\int }_a^{b}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛};}$
   ② ${\displaystyle \mathrm{\exists }\alpha \ge 1:\underset{x\to a+}{lim}(x-a{)}^{\alpha }f(x)=k(\ne 0)\mathrm{或}\mathrm{\infty }\Rightarrow {\int }_a^{b}f(x)\mathrm{d}x\mathrm{发}\mathrm{散}.}$
2. $f(x)\in c[a,b],b\mathrm{为}\mathrm{唯}\mathrm{一}\mathrm{瑕}\mathrm{点}(\mathrm{无}\mathrm{穷}\mathrm{间}\mathrm{断}\mathrm{点})$
   ① ${\displaystyle \mathrm{\exists }\alpha <1:\underset{x\to b-}{lim}(b-x{)}^{\alpha }f(x)\mathrm{\exists }\Rightarrow {\int }_a^{b}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛};}$
   ② ${\displaystyle \mathrm{\exists }\alpha \ge 1:\underset{x\to b-}{lim}(b-x{)}^{\alpha }f(x)=k(\ne 0)\mathrm{或}\mathrm{\infty }\Rightarrow {\int }_a^{b}f(x)\mathrm{d}x\mathrm{发}\mathrm{散}.}$
3. ${\displaystyle f(x)\in c[a,c)\cup (c,b]\mathrm{且}\underset{x\to c}{lim}f(x)=\mathrm{\infty }}$

Conc ${\displaystyle {\int }_a^{b}f(x)\mathrm{d}x\mathrm{收}\mathrm{敛}⟺{\int }_a^{c}f(x)\mathrm{d}x\mathrm{与}{\int }_c^{b}f(x)\mathrm{d}x\mathrm{都}\mathrm{收}\mathrm{敛}.}$

例1. ${\displaystyle {\int }_1^2\frac{1}{x\sqrt{x-1}}\mathrm{d}x}$
例2. ${\displaystyle {\int }_0^2\frac{1}{\sqrt{2x-x^2}}\mathrm{d}x}$