Definite Integrals / 定积分
Introduction / 介绍
Definite integrals are used to calculate the area under a curve over a specific interval. This cheat sheet summarizes key definitions, properties, and techniques for evaluating definite integrals, including those with infinite limits.
定积分用于计算在特定区间下曲线的面积。此备忘单总结了定积分的关键定义、性质和计算技巧，包括无穷界限的积分。
Definitions / 定义
Definite Integral / 定积分
The definite integral of a function from to is denoted by:
函数 从 到 的定积分表示为：
Fundamental Theorem of Calculus / 微积分基本定理
If is an antiderivative of , then:
如果 是 的一个原函数，则：
Properties / 性质
-
Linearity / 线性性质
\int_{a}^{c} f(x) , dx + \int_{c}^{b} f(x) , dx = \int_{a}^{b} f(x) , dx
\int_{a}^{b} f(x) , dx = -\int_{b}^{a} f(x) , dx
\int_{a}^{a} f(x) , dx = 0
## Techniques for Evaluation / 计算技巧 ### Basic Techniques / 基本技巧 1. **Substitution Method / 代换法** Change of variable: / 变量替换：\int_{a}^{b} f(g(x))g’(x) , dx = \int_{g(a)}^{g(b)} f(u) , du
\int_{a}^{b} u(x)v’(x) , dx = \left[ u(x)v(x) \right]{a}^{b} - \int{a}^{b} u’(x)v(x) , dx
where $u(x)$ and $v(x)$ are continuously differentiable functions on $[a, b]$. / 其中 $u(x)$ 和 $v(x)$ 在 $[a, b]$ 上是连续可微函数。 ### Special Techniques / 特殊技巧 1. **Improper Integrals / 广义积分** When the upper or lower limit (or both) is infinite, or the integrand becomes infinite within the interval of integration. 当积分的上限或下限（或两者）是无穷大，或者被积函数在积分区间内变为无穷大时。\int_{a}^{\infty} f(x) , dx = \lim_{b \to \infty} \int_{a}^{b} f(x) , dx
\int_{-\infty}^{b} f(x) , dx = \lim_{a \to -\infty} \int_{a}^{b} f(x) , dx
\int_{-\infty}^{\infty} f(x) , dx = \lim_{a \to -\infty, b \to \infty} \int_{a}^{b} f(x) , dx
## Examples / 示例 ### Basic Example / 基本示例 Evaluate $\int_{0}^{2} (3x^2 + 2x + 1) \, dx$: 计算 $\int_{0}^{2} (3x^2 + 2x + 1) \, dx$：\begin{align*}
\int_{0}^{2} (3x^2 + 2x + 1) , dx &= \left[ x^3 + x^2 + x \right]_{0}^{2} \
&= (2^3 + 2^2 + 2) - (0^3 + 0^2 + 0) \
&= 8 + 4 + 2 \
&= 14
\end{align*}
### Improper Integral Example / 广义积分示例 Evaluate $\int_{1}^{\infty} \frac{1}{x^2} \, dx$: 计算 $\int_{1}^{\infty} \frac{1}{x^2} \, dx$：\begin{align*}
\int_{1}^{\infty} \frac{1}{x^2} , dx &= \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^2} , dx \
&= \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_{1}^{b} \
&= \lim_{b \to \infty} \left( -\frac{1}{b} + \frac{1}{1} \right) \
&= 0 + 1 \
&= 1
\end{align*}
## Common Integrals / 常见积分 1. $\int_{a}^{b} k \, dx = k(b - a)$ 2. $\int_{a}^{b} x \, dx = \frac{1}{2} (b^2 - a^2)$ 3. $\int_{a}^{b} x^2 \, dx = \frac{1}{3} (b^3 - a^3)$ 4. $\int_{a}^{b} e^x \, dx = e^x \Big|_{a}^{b} = e^b - e^a$ 5. $\int_{a}^{b} \sin(x) \, dx = -\cos(x) \Big|_{a}^{b} = -\cos(b) + \cos(a)$ 6. $\int_{a}^{b} \cos(x) \, dx = \sin(x) \Big|_{a}^{b} = \sin(b) - \sin(a)$ --- ## Advanced Calculation Techniques / 进阶计算技巧 ### 1. Exchange of Limits and Integration / 积分与极限的交换 #### Concept / 概念 In some cases, it is possible to exchange the order of a limit and an integral. This is often used in more advanced calculus, particularly in series and sequences of functions. 在某些情况下，可以交换极限和积分的顺序。这在更高阶的微积分中尤为常见，特别是在函数的级数和序列中。 #### Theorem / 定理 If $f(x, t)$ is a function such that the following conditions are satisfied: - $f(x, t)$ is continuous in $x$ for each fixed $t$, - The limit $\lim_{t \to t_0} f(x, t)$ exists for each $x$ in the interval $[a, b]$, - There exists an integrable function $g(x)$ such that $|f(x, t)| \leq g(x)$ for all $t$ near $t_0$, then:\lim_{t \to t_0} \int_{a}^{b} f(x, t) , dx = \int_{a}^{b} \lim_{t \to t_0} f(x, t) , dx
如果函数 $f(x, t)$ 满足以下条件： - 对于每一个固定的 $t$， $f(x, t)$ 在 $x$ 上是连续的， - 对于区间 $[a, b]$ 上的每个 $x$，极限 $\lim_{t \to t_0} f(x, t)$ 存在， - 存在一个可积函数 $g(x)$，使得对于所有 $t$ 近 $t_0$ 时 $|f(x, t)| \leq g(x)$， 则有：\lim_{t \to t_0} \int_{a}^{b} f(x, t) , dx = \int_{a}^{b} \lim_{t \to t_0} f(x, t) , dx
#### Example / 示例 Evaluate $\lim_{n \to \infty} \int_{0}^{1} x^n \, dx$: 计算 $\lim_{n \to \infty} \int_{0}^{1} x^n \, dx$：\begin{align*}
\int_{0}^{1} x^n , dx &= \left[ \frac{x^{n+1}}{n+1} \right]_{0}^{1} \
&= \frac{1}{n+1}
\end{align*}
\lim_{n \to \infty} \int_{0}^{1} x^n , dx = \lim_{n \to \infty} \frac{1}{n+1} = 0
### 2. Differentiation Under the Integral Sign / 积分号下微分 #### Concept / 概念 This technique allows differentiation of an integral with respect to a parameter. 这一技巧允许我们对含参数积分的参数进行微分。 #### Theorem (Leibniz Integral Rule) / 定理（莱布尼兹积分法则） If $f(x, t)$ is continuous on $[a, b] \times [c, d]$ and has a continuous partial derivative with respect to $t$, then: 如果函数 $f(x, t)$ 在 $[a, b] \times [c, d]$ 上连续，并且它对 $t$ 的偏导数也是连续的，那么：\frac{d}{dt} \int_{a(t)}^{b(t)} f(x, t) , dx = f(b(t), t) \frac{db(t)}{dt} - f(a(t), t) \frac{da(t)}{dt} + \int_{a(t)}^{b(t)} \frac{\partial}{\partial t} f(x, t) , dx
#### Example / 示例 Evaluate $\frac{d}{dt} \int_{0}^{t} e^{-x^2} \, dx$: 计算 $\frac{d}{dt} \int_{0}^{t} e^{-x^2} \, dx$：\begin{align*}
\frac{d}{dt} \int_{0}^{t} e^{-x^2} , dx &= e^{-t^2} \cdot \frac{d}{dt}(t) - e^{-0^2} \cdot \frac{d}{dt}(0) + \int_{0}^{t} \frac{\partial}{\partial t} (e^{-x^2}) , dx \
&= e^{-t^2}
\end{align*}
### 3. Fubini's Theorem / 富比尼定理 #### Concept / 概念 Fubini's theorem provides conditions under which a double integral can be computed as an iterated integral. 富比尼定理提供了双重积分可以作为迭代积分计算的条件。 #### Theorem / 定理 If $f(x, y)$ is continuous on the rectangle $[a, b] \times [c, d]$, then: 如果函数 $f(x, y)$ 在矩形 $[a, b] \times [c, d]$ 上是连续的，那么：\int_{a}^{b} \int_{c}^{d} f(x, y) , dy , dx = \int_{c}^{d} \int_{a}^{b} f(x, y) , dx , dy
#### Example / 示例 Evaluate $\int_{0}^{1} \int_{0}^{1} xy \, dx \, dy$: 计算 $\int_{0}^{1} \int_{0}^{1} xy \, dx \, dy$：\begin{align*}
\int_{0}^{1} \int_{0}^{1} xy , dx , dy &= \int_{0}^{1} \left( \int_{0}^{1} xy , dx \right) dy \
&= \int_{0}^{1} \left( y \int_{0}^{1} x , dx \right) dy \
&= \int_{0}^{1} \left( y \cdot \frac{1}{2} \right) dy \
&= \frac{1}{2} \int_{0}^{1} y , dy \
&= \frac{1}{2} \cdot \frac{1}{2} \
&= \frac{1}{4}
\end{align*}