📚 Improper Integrals | 反常积分
An improper integral arises when an integral has an infinite limit of integration or when the integrand becomes unbounded within the interval of integration. These situations cannot be handled by the standard Riemann integral, so we define them through limits.
反常积分出现在积分区间包含无穷大，或者被积函数在积分区间内趋于无穷大的情形。这类积分无法用普通定积分直接处理，需要通过极限过程来定义。
1. What Are Improper Integrals? | 什么是反常积分？
There are two main types of improper integrals. The first type involves one or both limits of integration being infinite, for example ∫₁^∞ f(x) dx.
反常积分主要有两类。第一类是积分上限或下限为无穷大，例如∫₁^∞ f(x) dx。
The second type occurs when the integrand has a vertical asymptote inside or at the boundary of the integration interval, such as ∫₀^1 1/x dx.
第二类是被积函数在积分区间内部或端点处有垂直渐近线，例如∫₀^1 1/x dx。
In both cases we replace the problematic point by a variable limit and then take a limit.
在这两种情形中，我们都用可变极限代替问题点，然后再求极限。
2. Type 1: Infinite Limits | 第一类：无穷限积分
If f(x) is continuous for x ≥ a, we define the improper integral with an infinite upper limit as:
若f(x)在x ≥ a上连续，我们定义上限为无穷大的反常积分为：
∫ₐ^∞ f(x) dx = limt→∞ ∫ₐ^t f(x) dx
Similarly, for an infinite lower limit we write ∫_{−∞}^b f(x) dx = limt→−∞ ∫_t^b f(x) dx.
类似地，下限为负无穷时，定义∫_{−∞}^b f(x) dx = limt→−∞ ∫_t^b f(x) dx。
If both limits are infinite, we split the integral at any real number c, for example:
当上下限均为无穷时，可在任意实数c处拆分积分，例如：
∫_{−∞}^∞ f(x) dx = ∫_{−∞}^c f(x) dx + ∫_c^∞ f(x) dx
The original integral converges only if both parts converge.
原积分只有当两个部分都收敛时才算收敛。
3. Type 2: Discontinuous Integrands | 第二类：瑕积分
If f(x) is continuous on [a, b) but becomes unbounded as x approaches b from the left, we define:
若f(x)在[a, b)上连续，但当x从左侧趋于b时趋于无穷大，我们定义：
∫ₐ^b f(x) dx = limε→0⁺ ∫ₐ^{b−ε} f(x) dx
If the discontinuity occurs at the lower limit a, we use limε→0⁺ ∫_{a+ε}^b f(x) dx.
如果间断点在下限a，则使用limε→0⁺ ∫_{a+ε}^b f(x) dx。
If there is a discontinuity at an interior point c, the integral is split into two parts:
如果间断点在内点c，则积分需要拆分为两部分：
∫ₐ^b f(x) dx = ∫ₐ^c f(x) dx + ∫_c^b f(x) dx
Both parts must converge for the entire integral to converge.
两部分都必须收敛，整个积分才收敛。
4. Convergence and Divergence | 收敛与发散
We say an improper integral converges if the corresponding limit exists and is finite. Otherwise, it diverges.
如果所对应的极限存在且有限，则称反常积分收敛；否则称为发散。
For example, ∫₁^∞ 1/x² dx converges because limt→∞ (1 − 1/t) = 1.
例如，∫₁^∞ 1/x² dx收敛，因为limt→∞ (1 − 1/t) = 1。
On the other hand, ∫₁^∞ 1/x dx diverges because limt→∞ ln(t) = ∞.
另一方面，∫₁^∞ 1/x dx发散，因为limt→∞ ln(t) = ∞。
Similarly, ∫₀^1 1/√x dx converges to 2, while ∫₀^1 1/x dx diverges.
同样，∫₀^1 1/√x dx收敛于2，而∫₀^1 1/x dx发散。
5. Evaluating Infinite Integrals | 计算无穷限积分
To evaluate an improper integral with an infinite limit, first compute the definite integral with a finite variable limit, then take the limit.
计算无穷限反常积分时，先计算以有限变量为上限的定积分，再求极限。
Example: evaluate ∫₁^∞ 1/x³ dx.
例：计算∫₁^∞ 1/x³ dx。
First, ∫₁^t 1/x³ dx = [−1/(2x²)]₁^t = −1/(2t²) + 1/2.
首先，∫₁^t 1/x³ dx = [−1/(2x²)]₁^t = −1/(2t²) + 1/2。
Taking the limit as t → ∞ gives 1/2, so the integral converges to 1/2.
令t → ∞取极限得到1/2，因此该积分收敛于1/2。
When the integrand includes an exponential, we often use the fact that limt→∞ t e^{−t} = 0.
当被积函数包含指数函数时，我们常利用limt→∞ t e^{−t} = 0这一事实。
6. Evaluating Integrals with Discontinuities | 计算含间断点的积分
For a discontinuity at the endpoint, we use a one-sided limit.
对于端点的间断点，我们使用单侧极限。
Example: evaluate ∫₀^1 1/√(1−x) dx.
例：计算∫₀^1 1/√(1−x) dx。
Since the integrand is unbounded at x = 1, set u = 1 − x. Then ∫₀^{1−ε} (1−x)^{−1/2} dx = [−2√(1−x)]₀^{1−ε} = 2 − 2√ε.
因为被积函数在x = 1处无界，令u = 1 − x，则∫₀^{1−ε} (1−x)^{−1/2} dx = [−2√(1−x)]₀^{1−ε} = 2 − 2√ε。
Taking limε→0⁺ gives 2, so the integral converges to 2.
取limε→0⁺得到2，因此积分收敛于2。
For an interior discontinuity, evaluate the two pieces separately and add them only if both limits exist.
对于内部间断点，分别计算两个部分，只有当两个极限都存在时才能相加。
7. Comparison Test | 比较判别法
The comparison test is useful when the integrand is difficult to integrate directly.
当被积函数难以直接积分时，比较判别法非常有用。
Suppose 0 ≤ f(x) ≤ g(x) for all x in the region of integration.
假设在积分区域内0 ≤ f(x) ≤ g(x)。
If ∫ g(x) dx converges, then ∫ f(x) dx also converges.
若∫ g(x) dx收敛，则∫ f(x) dx也收敛。
If ∫ f(x) dx diverges, then ∫ g(x) dx also diverges.
若∫ f(x) dx发散，则∫ g(x) dx也发散。
Example: determine whether ∫₁^∞ (sin²x)/x² dx converges.
例：判断∫₁^∞ (sin²x)/x² dx是否收敛。
Since 0 ≤ (sin²x)/x² ≤ 1/x² and ∫₁^∞ 1/x² dx converges, the given integral converges.
因为0 ≤ (sin²x)/x² ≤ 1/x²，且∫₁^∞ 1/x² dx收敛，所以该积分收敛。
8. Standard Results | 常见标准结论
The p-integral is a classic standard result.
p-积分是一个经典的标准结论。
∫₁^∞ 1/xᵖ dx 收敛当且仅当 p > 1
∫₀^1 1/xᵖ dx 收敛当且仅当 p < 1
Another useful result is the exponential integral:
另一个有用的结论是指数积分：
∫₀^∞ e^{−kx} dx = 1/k , for k > 0
Also remember that the Gaussian integral ∫_{−∞}^∞ e^{−x²} dx = √π is widely used in probability and statistics.
另外，高斯积分∫_{−∞}^∞ e^{−x²} dx = √π在概率和统计中广泛使用。
9. Applications | 实际应用
Improper integrals appear naturally in probability density functions, where the total area under a density curve must be 1.
反常积分自然出现在概率密度函数中，密度曲线下的总面积为1。
For an exponential distribution with rate λ, the density is f(x) = λ e^{−λx} for x ≥ 0. The total probability is ∫₀^∞ λ e^{−λx} dx = 1.
对于速率为λ的指数分布，密度为f(x) = λ e^{−λx}（x ≥ 0）。总概率为∫₀^∞ λ e^{−λx} dx = 1。
In physics, improper integrals are used to compute work done in moving an object to infinity against a force, such as gravitational potential energy.
在物理学中，反常积分用于计算将物体反抗力移动到无穷远所做的功，例如引力势能。
In engineering, Laplace transforms are defined as improper integrals and are used to solve differential equations.
在工程中，拉普拉斯变换就是反常积分，常用于求解微分方程。
10. Common Pitfalls | 常见错误
One common mistake is ignoring discontinuities inside the interval. Always check the integrand for vertical asymptotes before evaluating.
常见错误之一是忽略区间内部的间断点。在计算前一定要检查被积函数是否存在垂直渐近线。
Another mistake is applying the comparison test in the wrong direction. If the larger integral diverges, the smaller one can still converge.
另一个错误是错误地使用比较判别法。若较大的积分发散，较小的积分仍可能收敛。
Also remember that ∫_{−∞}^∞ f(x) dx is not simply limt→∞ ∫_{−t}^t f(x) dx unless the integral converges symmetrically in the Cauchy principal value sense. For an improper integral, both tails must converge separately.
还要注意，∫_{−∞}^∞ f(x) dx并不能简单地写成limt→∞ ∫_{−t}^t f(x) dx，除非是柯西主值意义下。对于反常积分，两个尾部必须分别收敛。
Finally, never treat ∞ as a number. Always use limits.
最后，不要把∞当作一个数，务必使用极限。
11. Practice and Summary | 练习与总结
Practice by identifying which type of improper integral is present, then rewrite it as a limit and evaluate.
练习时先判断属于哪类反常积分，然后改写为极限并计算。
Try these examples: (a) ∫₂^∞ 1/(x ln²x) dx; (b) ∫₀^1 ln x dx; (c) ∫₀^∞ x e^{−x²} dx.
尝试这些例子：(a) ∫₂^∞ 1/(x ln²x) dx；(b) ∫₀^1 ln x dx；(c) ∫₀^∞ x e^{−x²} dx。
Summary: improper integrals require limits; convergence means the limit is finite; always handle infinite limits and discontinuities separately; use comparison test when direct evaluation is hard.
总结：反常积分需要极限；收敛意味着极限有限；务必分别处理无穷限和间断点；当直接计算困难时使用比较判别法。
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