Integration (Pure 2) · ⁨积分(Pure 2)⁩ — A-Level Mathematics

Vocabulary · ⁨词汇⁩

English	中文	Pinyin · ⁨拼音⁩
integral/ˈɪntɪɡrəl/	积分	jī fēn
exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/	指数函数	zhǐ shù hán shù
differential equation/ˌdɪfəˈrenʃl ɪˈkweɪʒn/	微分方程	wēi fēn fāng chéng
trapezium rule/trəˈpiːzɪəm ruːl/	梯形法则	tī xíng fǎ zé
strip/strɪp/	条带	tiáo dài
estimate/ˈestɪmət/	估计	gū jì
identity/aɪˈdentɪti/	恒等式	héng děng shì

English

The area that eˣ can't escape

The integral 积分 of $e^x$ is $e^x$ — the same function. This self-referential property makes exponential functions 指数函数 central to differential equations 微分方程.
Integration techniques let you find areas, volumes, and solutions to real-world problems.

中文

eˣ 逃不掉的面积

$e^x$ 的积分是 $e^x$——同一个函数。这个自我指涉的性质使指数函数成为微分方程的核心。
积分技巧让你找到面积、体积和现实问题的解。

English

Standard integrals

$$\int e^{ax+b}\,dx = \frac{1}{a}e^{ax+b} + C, \qquad \int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln|ax+b| + C,$$

$$\int \cos(ax+b)\,dx = \frac{1}{a}\sin(ax+b) + C, \qquad \int \sin(ax+b)\,dx = -\frac{1}{a}\cos(ax+b) + C.$$

Example

Worked example. $\int e^{3x}\,dx = \dfrac{1}{3}e^{3x} + C$. The $\dfrac{1}{3}$ comes from the chain rule in reverse.

Watch out

Don't forget the $\dfrac{1}{a}$. When integrating $e^{ax+b}$, you must divide by $a$. The integral of $e^{2x}$ is $\dfrac{1}{2}e^{2x}$, not $e^{2x}$.

The trapezium rule 梯形法则 adds straight-topped strips 条带 of width h to estimate 估计 the area

中文

标准积分

$$\int e^{ax+b}\,dx = \frac{1}{a}e^{ax+b} + C, \qquad \int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln|ax+b| + C,$$

$$\int \cos(ax+b)\,dx = \frac{1}{a}\sin(ax+b) + C, \qquad \int \sin(ax+b)\,dx = -\frac{1}{a}\cos(ax+b) + C.$$

例子

算例。 $\int e^{3x}\,dx = \dfrac{1}{3}e^{3x} + C$。那个 $\dfrac{1}{3}$ 来自反过来的链式法则。

注意

不要忘记 $\dfrac{1}{a}$。 当积分 $e^{ax+b}$ 时,你必须除以 $a$。$e^{2x}$ 的积分是 $\dfrac{1}{2}e^{2x}$,不是 $e^{2x}$。

梯形法则把宽度为 h 的平顶条带加起来估计面积

Explore · ⁨探索⁩

The area under the curve · ⁨曲线下的面积⁩

area = ∫ f(x) dx

The integral still measures area · ⁨面积⁩ — drag a and b to total the strip under the curve. · ⁨积分仍然测量面积——拖动 a 和 b 来合计曲线下的条带。⁩

Practice · ⁨练习⁩

What is ∫ eˣ dx? · ⁨∫ eˣ dx 是什么?⁩

eˣ + C
x eˣ + C
eˣ⁺¹ + C
ln x + C

Practice · ⁨练习⁩

What is ∫ (1/x) dx? · ⁨∫ (1/x) dx 是什么?⁩

ln|x| + C
1/x² + C
x⁻² + C
eˣ + C

Practice · ⁨练习⁩

∫ e^(2x) dx = (1/a)e^(2x) + C. What is a? · ⁨∫ e^(2x) dx = (1/a)e^(2x) + C。a 是多少?⁩

Practice · ⁨练习⁩

∫ cos(3x) dx = (1/3)sin(3x) + C. Evaluate from 0 to π/6 (2 dp). · ⁨∫ cos(3x) dx = (1/3)sin(3x) + C。从 0 到 π/6 求值(2 位小数)。⁩

English

Integrating powers of trig functions

To integrate a power of $\sin$/$\cos$, use an identity 恒等式 to remove the power first.
Example: $\sin^2 x = \dfrac{1}{2}(1 - \cos 2x)$, so $\int \sin^2 x\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$.

中文

积分三角函数的幂

要积分 $\sin$/$\cos$ 的一个幂,先用一个恒等式去掉幂。
例子:$\sin^2 x = \dfrac{1}{2}(1 - \cos 2x)$,所以 $\int \sin^2 x\,dx = \dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$。

English

The trapezium rule

When you can't integrate exactly, estimate:

$$\int_a^b y\,dx \approx \tfrac{h}{2}\big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\big]$$

Each strip of width $h$ is a trapezium; add their areas.

The trapezium rule: divide the area into strips, approximate each as a trapezium, and sum their areas.

中文

梯形法则

当你不能精确积分时,估计:

$$\int_a^b y\,dx \approx \tfrac{h}{2}\big[y_0 + y_n + 2(y_1 + \cdots + y_{n-1})\big]$$

每个宽度 $h$ 的条带是一个梯形;把它们的面积相加。

梯形法则:把面积分成条带,把每个近似为一个梯形,并把它们的面积求和。

Practice · ⁨练习⁩

The trapezium rule estimates a definite integral when it cannot be found exactly. · ⁨当一个定积分不能被精确求出时,梯形法则估计它。⁩

Practice · ⁨练习⁩

Using the trapezium rule with h = 1, estimate ∫₁³ x² dx with y-values 1, 4, 9. · ⁨用 h = 1 的梯形法则,用 y 值 1、4、9 估计 ∫₁³ x² dx。⁩

English

Worked example — trapezium rule

Estimate $\int_1^4 x^2\,dx$ using 3 strips ($h = 1$).
$y$-values: $y_0 = 1$, $y_1 = 4$, $y_2 = 9$, $y_3 = 16$.
$\int \approx \dfrac{1}{2}[1 + 16 + 2(4 + 9)] = \dfrac{1}{2}[17 + 26] = 21.5$.
Exact value: $\left[\dfrac{x^3}{3}\right]_1^4 = \dfrac{64}{3} - \dfrac{1}{3} = 21$. The estimate is close.
Integration is reverse differentiation; harder integrals use integration by substitution.

中文

算例——梯形法则

用 3 个条带($h = 1$)估计 $\int_1^4 x^2\,dx$。
$y$ 值:$y_0 = 1$,$y_1 = 4$,$y_2 = 9$,$y_3 = 16$。
$\int \approx \dfrac{1}{2}[1 + 16 + 2(4 + 9)] = \dfrac{1}{2}[17 + 26] = 21.5$。
精确值:$\left[\dfrac{x^3}{3}\right]_1^4 = \dfrac{64}{3} - \dfrac{1}{3} = 21$。估计很接近。
积分是反向求导(reverse differentiation);较难的积分用换元积分(integration by substitution)。

English

You've got it

Key idea

$\int e^{ax+b}dx = \tfrac1a e^{ax+b} + C$; $\int \tfrac{1}{ax+b}dx = \tfrac1a\ln|ax+b| + C$
remove powers of $\sin$/$\cos$ with an identity before integrating
the trapezium rule estimates an integral as a sum of trapezium strips

中文