Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation · 寻找原函数与不定积分：基本规则与记号
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| indefinite integral/ɪnˈdefɪnət ˈɪntɪɡrəl/ | 不定积分 | bù dìng jī fēn |
| constant of integration/ˈkɒnstənt ɒv ˌɪntɪˈɡreɪʃn/ | 积分常数 | jī fēn cháng shù |
Running differentiation backward
- FTC Part 2 needs an antiderivative — so let's learn to find them.
- An antiderivative of $f$ is any function whose derivative is $f$; the whole family is the indefinite integral 不定积分.
- $\displaystyle\int f(x)\,dx$ (no limits) means "the general antiderivative of $f$."
- Every derivative rule, read backward, becomes an antiderivative rule.
把求导倒着运行
- FTC 第二部分需要一个原函数——那就来学怎么找它们。
- $f$ 的原函数是任何导数为 $f$ 的函数;它的整个族就是不定积分。
- $\displaystyle\int f(x)\,dx$(无积分限)表示"$f$ 的一般原函数"。
- 每条求导规则倒着读,就变成一条原函数规则。
The power rule for integration
- Reverse the Power Rule: add one to the exponent, then divide by the new exponent.
-
$$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\qquad(n\neq-1)$$
- Check: differentiating $\frac{x^{n+1}}{n+1}$ gives $x^n$. ✓
- (The case $n=-1$ is special: $\int x^{-1}\,dx=\ln|x|+C$.)
积分的幂法则
- 反转幂法则:指数加一,再除以新指数。
-
$$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\qquad(n\neq-1)$$
- 检验:对 $\frac{x^{n+1}}{n+1}$ 求导得 $x^n$。✓
- ($n=-1$ 的情形特殊:$\int x^{-1}\,dx=\ln|x|+C$。)
What is $\displaystyle\int x^3\,dx$? · $\displaystyle\int x^3\,dx$是什么？
Add one to the exponent, divide by it: $\tfrac{x^4}{4}+C$. · 指数加一，再除以其新值：$\tfrac{x^4}{4}+C$。
The power rule fails for $n=-1$. What is $\displaystyle\int x^{-1}\,dx$? · 幂法则不适用于 $n=-1$。$\displaystyle\int x^{-1}\,dx$ 是什么？
$\int\tfrac1x\,dx=\ln|x|+C$.
Never forget $+C$
- Since the derivative of a constant is $0$, any constant can be added — so the antiderivative is a family.
- Always append the constant of integration 积分常数 $+C$ to an indefinite integral.
- Without limits there's no way to pin down which constant, so $+C$ stays.
- (In a definite integral the $+C$ cancels — that's the difference.)
永远别忘 $+C$
- 因为常数的导数是 $0$,任何常数都能加上——所以原函数是一个族。
- 永远给不定积分附上积分常数 $+C$。
- 没有积分限就无法确定是哪个常数,所以 $+C$ 保留。
- (在定积分中 $+C$ 抵消——这就是区别。)
A family shifted by +C · 平移+C族
y = ax³ + d
All antiderivatives of $f$ differ only by a vertical shift $+C$ — same slope everywhere. · $f$ 的所有原函数仅相差一个垂直平移 $+C$——处处斜率相同。
An indefinite integral should include the constant of integration $+C$. · 不定积分应包含积分常数 $+C$。
The antiderivative is a family; $+C$ is part of the answer. · 原函数是一个族；$+C$ 是答案的一部分。
The standard antiderivatives
- Reverse the elementary derivatives you know:
- $\displaystyle\int\cos x\,dx=\sin x+C$; $\;\displaystyle\int\sin x\,dx=-\cos x+C$.
- $\displaystyle\int e^x\,dx=e^x+C$; $\;\displaystyle\int\frac1x\,dx=\ln|x|+C$.
- Combine with linearity to integrate any polynomial or sum term by term.
标准原函数
- 反转你已知的基本导数:
- $\displaystyle\int\cos x\,dx=\sin x+C$;$\;\displaystyle\int\sin x\,dx=-\cos x+C$。
- $\displaystyle\int e^x\,dx=e^x+C$;$\;\displaystyle\int\frac1x\,dx=\ln|x|+C$。
- 与线性性结合,逐项对任何多项式或和积分。
What is $\displaystyle\int \sin x\,dx$? · $\displaystyle\int \sin x\,dx$是什么？
$\int\sin x\,dx=-\cos x+C$ (mind the minus). · $\int\sin x\,dx=-\cos x+C$（注意负号）。
Find $\displaystyle\int (6x^2-4x+5)\,dx$. · 求 $\displaystyle\int (6x^2-4x+5)\,dx$。
Reverse power rule term by term, plus $C$. · 逐项反向使用幂法则，再加上 $C$。
Select all · 所有 correct antiderivatives. · 选择所有正确的原函数。
$\int\cos x\,dx=\sin x+C$ (positive); the last is wrong. · $\int\cos x\,dx=\sin x+C$（正号）；最后一项是错误的。
Two staples: always add $+C$ to an indefinite integral (it's part of the answer), and the power rule fails for $n=-1$ — $\int x^{-1}\,dx$ is $\ln|x|+C$, not $\frac{x^0}{0}$. Also mind the sign on $\int\sin x\,dx=-\cos x+C$ (a minus, mirroring $\cos$'s derivative).
两个要点:永远给不定积分加 $+C$(它是答案的一部分),而幂法则对 $n=-1$ 失效——$\int x^{-1}\,dx$ 是 $\ln|x|+C$,不是 $\frac{x^0}{0}$。也注意 $\int\sin x\,dx=-\cos x+C$ 的符号(一个负号,呼应 $\cos$ 的导数)。
Find $\displaystyle\int (6x^2 - 4x + 5)\,dx$.
- Term by term (reverse power rule): $6\cdot\dfrac{x^3}{3}=2x^3$; $\;-4\cdot\dfrac{x^2}{2}=-2x^2$; $\;5x$.
- $\displaystyle\int (6x^2-4x+5)\,dx = 2x^3-2x^2+5x+C$.
- Check by differentiating: $6x^2-4x+5$. ✓
求 $\displaystyle\int (6x^2 - 4x + 5)\,dx$。
- 逐项(反向幂法则):$6\cdot\dfrac{x^3}{3}=2x^3$;$\;-4\cdot\dfrac{x^2}{2}=-2x^2$;$\;5x$。
- $\displaystyle\int (6x^2-4x+5)\,dx = 2x^3-2x^2+5x+C$。
- 通过求导检验:$6x^2-4x+5$。✓
An indefinite integral $\int f\,dx$ is the general antiderivative of $f$. The power rule for integration: $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq-1$ (and $\int x^{-1}\,dx=\ln|x|+C$). Always add the constant of integration $+C$, and integrate sums term by term.
不定积分 $\int f\,dx$ 是 $f$ 的一般原函数。积分的幂法则:当 $n\neq-1$ 时 $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$(且 $\int x^{-1}\,dx=\ln|x|+C$)。永远加上积分常数 $+C$,并逐项对和积分。