Review set 20A - NON-CALCULATOR | 复习套题20A——非计算器

📚 Review set 20A - NON-CALCULATOR | 复习套题20A——非计算器

Review set 20A is a focused non‑calculator drill designed to strengthen your integration skills for the IB Mathematics Analysis and Approaches course. It covers indefinite and definite integrals, the reverse chain rule with linear inner functions, and area calculations between curves and the x‑axis. Working without a calculator forces you to rely on algebraic manipulation, standard antiderivative forms, and careful arithmetic — exactly the skills that are tested in Paper 1.

复习套题20A是一套针对IB数学分析与方法的非计算器专项训练，旨在强化你的积分技能。它涵盖了不定积分、定积分、线性内嵌函数的逆向链式法则，以及曲线与x轴之间面积的计算。在不使用计算器的条件下，你必须依赖代数变形、标准原函数形式和精确的算术运算——这些正是试卷一中考察的核心能力。

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1. Overview and Exam Focus | 概述与考试重点

In a non‑calculator IB exam, integration questions typically require you to recognise basic antiderivatives, apply linear substitutions, evaluate definite integrals using the Fundamental Theorem of Calculus, and find areas by integrating with appropriate limits. You must be confident working with fractions, negative exponents, and trigonometric values manually.

在非计算器IB考试中，积分题通常要求你识别基本原函数、应用线性换元、利用微积分基本定理计算定积分，并通过适当上下限求面积。你必须对手算分数、负指数和三角函数值充满信心。

The most common function families you will integrate are power functions of the form xⁿ, trigonometric functions such as sin x and cos x, exponential functions eˣ, and their composed linear forms like (ax + b)ⁿ, sin(ax + b), and eᵃˣ⁺ᵇ. You will also need to simplify integrands by expanding brackets or splitting fractions before integrating.

你需要积分的最常见函数族包括形如xⁿ的幂函数、sin x和cos x等三角函数、指数函数eˣ，以及它们的线性复合形式，例如(ax + b)ⁿ、sin(ax + b)和eᵃˣ⁺ᵇ。你还需在积分前通过展开括号或拆分分式来化简被积函数。

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2. Antiderivatives of Power Functions | 幂函数的不定积分

he power is a fraction, such as ½, write the result with a rational exponent or as a surd.

幂函数的积分法则指出，对于任意实数n ≠ −1，xⁿ的原函数为(xⁿ⁺¹)/(n + 1)再加积分常数。当指数为负时，需要小心处理分式；例如∫ x⁻² dx = −x⁻¹ + C。如果指数是分数，如½，结果可用有理指数或根式表示。

A typical non‑calculator question might ask you to find ∫ (2x³ − 4x + 1/√x) dx. First rewrite 1/√x s x⁻¹/², then integrate term by term: (2x⁴)/4 − (4x²)/2 + (x¹/²)/(1/2) + C = ½x⁴ − 2x² + 2√x + C. Always check that you have increased the exponent by one and divided by the new exponent.

一个典型的非计算器题目可能要求你求∫ (2x³ − 4x + 1/√x) dx。首先将1/√x改写为x⁻¹/²，然后逐项积分：(2x⁴)/4 − (4x²)/2 + (x¹/²)/(1/2) + C = ½x⁴ − 2x² + 2√x + C。务必确认指数加一后除以新指数。

∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C, n ≠ −1

∫ xⁿ dx = (xⁿ⁺¹)/(n + 1) + C，n ≠ −1

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3. Integrating Trigonometric Functions | 三角函数的积分

(ax + b) + C. The same pattern holds for cosine: ∫ cos(ax + b) dx = (1/a) sin(ax + b) + C.

正弦和余弦的标准积分很容易记忆：∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。当参数是线性函数时，例如sin(3x + 1)，应使用逆向链式法则：∫ sin(ax + b) dx = −(1/a) cos(ax + b) + C。余弦同理：∫ cos(ax + b) dx = (1/a) sin(ax + b) + C。

For non‑ ues at key angles (0, π/6, π/4, π/3, π/2) to evaluate definite integrals. For instance, ∫₀^π/2 cos x dx = [sin x]₀^π/2 = sin(π/2) − sin 0 = 1 − 0 = 1. Being fluent with radian measure is essential.

在非计算器运算中，你必须熟知关键角度（0、π/6、π/4、π/3、π/2）的精确三角函数值，以计算定积分。例如，∫₀^π/2 cos x dx = [sin x]₀^π/2 = sin(π/2) − sin 0 = 1 − 0 = 1。熟练使用弧度制至关重要。

Function	Antiderivative
sin(ax + b)	−(1/a) cos(ax + b) + C
cos(ax + b)	(1/a) sin(ax + b) + C

表格：函数 sin(ax + b) 的原函数为 −(1/a) cos(ax + b) + C；cos(ax + b) 的原函数为 (1/a) sin(ax + b) + C。

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4. Exponential and Logarithmic Integration | 指数与对数函数的积分

The exponential function eˣ is its own antiderivative: ∫ eˣ dx = eˣ + C. For a linear exponent, ∫ eᵃˣ⁺ᵇ dx = (1/a) eᵃˣ⁺ᵇ + C. For example, ∫ e⁵ˣ⁻² dx = (1/5) e⁵ˣ⁻² + C. When the base is not e, such as aˣ, the integral is aˣ / ln a + C, but such questions are rare in non‑calculator papers because ln a may not be a nice number.

指数函数eˣ的导数即其本身，因此∫ eˣ dx = eˣ + C。对于线性指数，∫ eᵃˣ⁺ᵇ dx = (1/a) eᵃˣ⁺ᵇ + C。例如，∫ e⁵ˣ⁻² dx = (1/5) e⁵ˣ⁻² + C。当底数不是e时，如aˣ，其积分为aˣ / ln a + C，但此类问题在非计算器试卷中很少出现，因为ln a可能不是整数。

Logarithmic integration arises from the derivative of ln x: ∫ (1/x) dx = ln |x| + C. More generally, ∫ (f'(x)/f(x)) dx = ln |f(x)| + C. A common non‑ t the numerator is the derivative of the denominator, for instance ∫ (2x/(x² + 5)) dx = ln |x² + 5| + C. Linear forms like ∫ 1/(3x + 2) dx = (1/3) ln |3x + 2| + C also appear.

对数积分源于ln x的导数：∫ (1/x) dx = ln |x| + C。更一般地，∫ (f'(x)/f(x)) dx = ln |f(x)| + C。一个常见的非计算器任务是识别分子是否为分母的导数，例如∫ (2x/(x² + 5)) dx = ln |x² + 5| + C。诸如∫ 1/(3x + 2) dx = (1/3) ln |3x + 2| + C的线性形式也会出现。

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5. Reverse Chain Rule (Linear Substitutions) | 逆向链式法则（线性换元）

F(ax + b) + C. This technique works for powers, trigonometric functions, and exponentials.

当被积函数是内嵌线性函数的复合函数时，可通过逆向链式法则积分。若已知∫ f(u) du = F(u) + C，则∫ f(ax + b) dx = (1/a) F(ax + b) + C。该技巧适用于幂函数、三角函数和指数函数。

⁸ / 8 ) + C = (4x − 1)⁸ / 32 + C. Always divide by the coefficient of x inside the bracket — a common slip is to multiply instead.

例如，计算∫ (4x − 1)⁷ dx时，将f(u)视作u⁷，其中u = 4x − 1。u⁷的原函数为u⁸/8，故答案为(1/4) · ( (4x − 1)⁸ / 8 ) + C = (4x − 1)⁸ / 32 + C。务必除以括号内x的系数——常见的失误是乘法与除法混淆。

a x outside. You would set u = x² + 1, du = 2x dx, and the integral becomes ½ ∫ u⁵ du. Non‑ s usually keep the numbers simple so that division yields whole numbers or easy fractions.

当内嵌函数不是线性时需要小心，例如∫ x(x² + 1)⁵ dx。此时x² + 1的导数为2x，恰与外面的x成比例。可设u = x² + 1，du = 2x dx，积分化为½ ∫ u⁵ du。非计算器试卷通常使数字简单，以便除法得到整数或简单分数。

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6. Definite Integrals Without Technology | 不使用计算器的定积分

(x) dx by hand, first find an antiderivative F(x), then compute F(b) − F(a). Simplify the expression step by step, paying close attention to signs and fractions. Many marks are lost through arithmetic errors when substituting limits.

手算定积分∫ₐᵇ f(x) dx时，先求出原函数F(x)，再计算F(b) − F(a)。逐步化简表达式，特别注意符号和分数。很多失分是由于代入上下限时的算术错误。

s 12 − (−6) = 18. Always write the limits clearly and use brackets to avoid sign mistakes.

考虑∫₋₁² (3x² − 2x + 4) dx。其原函数为x³ − x² + 4x。代入上限：(8 − 4 + 8) = 12。代入下限：(−1 − 1 − 4) = −6。结果为12 − (−6) = 18。务必清晰标注上下限并使用括号以避免符号错误。

trigonometric functions, remember to evaluate the sine and cosine exactly: for example, ∫₀^π/3 sin(2x) dx = [−½ cos(2x)]₀^π/3 ) = −½(−½) + ½(1) = ¼ + ½ = ¾. The ability to recall cos(2π/3) = −½ by heart is critical.

当被积函数包含三角函数时，记住要精确计算正弦和余弦值：例如∫₀^π/3 sin(2x) dx = [−½ cos(2x)]₀^π/3 = −½ cos(2π/3) − ( −½ cos 0 ) = −½(−½) + ½(1) = ¼ + ½ = ¾。能否心算出cos(2π/3) = −½十分关键。

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7. Area Between a Curve and the x‑Axis | 曲线与x轴之间的面积

The area bounded by y = f(x), the x‑axis, and the vertical lines x = a and x = b is given by ∫ₐᵇ |f(x)| dx. When a graph crosses the x‑ arately. Without a calculator, you need to solve f(x) = 0 to find the intercepts manually.

由y = f(x)、x轴以及直线x = a和x = b围成的面积等于∫ₐᵇ |f(x)| dx。当图像在区间内穿过x轴时，必须将积分拆分为f(x)为正和为负的区域，分别取各部分的绝对值。没有计算器时，你需要手动解方程f(x) = 0来找到截点。

For instance, find the total area enclosed between y = x² − 4x + 3 and the x‑ , then add A₁ + A₂ + A₃. Work each integral out carefully to obtain the exact rational result.

例如，求y = x² − 4x + 3与x轴之间从x = 0到x = 4的总面积。首先因式分解：y = (x − 1)(x − 3)。曲线在x = 1和x = 3处与x轴相交。在[0,1]上函数为正，在[1,3]上为负，在[3,4]上又为正。分别计算A₁ = ∫₀¹ (x² − 4x + 3) dx，A₂ = −∫₁³ (x² − 4x + 3) dx，A₃ = ∫₃⁴ (x² − 4x + 3) dx，再将A₁ + A₂ + A₃相加。仔细计算每个积分，得出精确的有理数结果。

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8. Area Between Two Curves | 两条曲线之间的面积

a enclosed by two curves y = f(x) and y = g(x) between x = a and x = b, the formula is ∫ₐᵇ |f(x) − g(x)| (x) = g(x) algebraically to find intersection points and set up the integrals accordingly.

当需要求两条曲线y = f(x)和y = g(x)在x = a与x = b之间所围面积时，公式为∫ₐᵇ |f(x) − g(x)| dx。通过在子区间内测试一个点或绘制简图来确定哪条函数在上方。没有计算器时，必须用代数方法解f(x) = g(x)求出交点，并据此建立积分。

s such as y = x² + 1 and y = 5 − x². Equating them gives x² + 1 = 5 − x² → 2x² = 4 → x = ±√2. The area between them from −√2 to √2 is ∫₋√₂^√2 [ (5 − x²) − (x² + 1) ] dx = ∫₋√₂^√2 (4 − 2x²) dx. The integrand is even, so you can double the integral from 0 to √2, simplifying arithmetic: 2 ∫₀^√2 (4 − 2x²) dx = 2 [4x − (2/3)x³]₀^√2 = 2 (4√2 − (2/3)·2√2 ) = 2 (4√2 − (4√2)/3) = 2 · (8√2/3) = (16√2)/3 square units.

一个典型题目给出曲线y = x² + 1和y = 5 − x²。令其相等得x² + 1 = 5 − x² → 2x² = 4 → x = ±√2。从−√2到√2之间的面积是∫₋√₂^√2 [ (5 − x²) − (x² + 1) ] dx = ∫₋√₂^√2 (4 − 2x²) dx。被积函数为偶函数，因此可将0到√2的积分乘以2以简化运算：2 ∫₀^√2 (4 − 2x²) dx = 2 [4x − (2/3)x³]₀^√2 = 2 (4√2 − (2/3)·2√2 ) = 2 (4√2 − (4√2)/3) = 2 · (8√2/3) = (16√2)/3 平方单位。

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9. Mixed Practice Problems with Solutions | 混合练习题及解答

Let's work through three non‑calculator problems that mirror the style of Review set 20A. These combine power, trigonometric, and exponential integration with area application.

让我们演练三道非计算器题目，它们反映了复习套题20A的风格。这些题综合了幂函数、三角函数和指数函数的积分以及面积应用。

Problem 1: Find ∫ (6x² − cos(4x) + 2e⁻ˣ) dx.

问题1：求∫ (6x² − cos(4x) + 2e⁻ˣ) dx。

Integrate term by term: 6x² gives 6 · (x³/3) = 2x³. −cos(4x) gives −(1/4) sin(4x). 2e⁻ˣ gives 2 · (1/(−1)) e⁻ˣ = −2e⁻ˣ. Don't forget the constant: the answer is 2x³ − (1/4) sin(4x) − 2e⁻ˣ + C.

逐项积分：6x²给出6 · (x³/3) = 2x³。−cos(4x)给出−(1/4) sin(4x)。2e⁻ˣ给出2 · (1/(−1)) e⁻ˣ = −2e⁻ˣ。别忘了常数：答案为2x³ − (1/4) sin(4x) − 2e⁻ˣ + C。

Problem 2: Evaluate ∫₁⁴ (2x + 1)/√x dx.

问题2：计算∫₁⁴ (2x + 1)/√x dx。

2 = 32/3 + 4 = 44/3. At x=1, (4/3)·1 + 2·1 = 4/3 + 2 = 10/3. Subtract: 44/3 − 10/3 = 34/3.

改写为∫ (2x¹/² + x⁻¹/²) dx。原函数为2 · (x³/²)/(3/2) + (x¹/²)/(1/2) = (4/3) x³/² + 2x¹/²。代入上下限：x=4时，(4/3)·8 + 2·2 = 32/3 + 4 = 44/3。x=1时，(4/3)·1 + 2·1 = 4/3 + 2 = 10/3。相减得44/3 − 10/3 = 34/3。

Problem 3: nd y = 4x − x² intersect at x = 0 and x = 3. Find the area of the region enclosed by them.

问题3：曲线y = x² − 2x与y = 4x − x²相交于x = 0和x = 3。求它们所围区域的面积。

2x) dx = ∫₀³ (6x − 2x²) dx. Integrate: [3x² − (2/3)x³]₀³ = (27 − 18) − 0 = 9 square units.

判断上下位置：在x = 1处，y分别为−1和3，故4x − x²为上曲线。面积为∫₀³ [ (4x − x²) − (x² − 2x) ] dx = ∫₀³ (4x − x² − x² + 2x) dx = ∫₀³ (6x − 2x²) dx。积分为[3x² − (2/3)x³]₀³ = (27 − 18) − 0 = 9 平方单位。

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10. Common Mistakes and Tips | 常见错误与技巧

nts incorrectly write (3x+2)⁵/5. Remember: the derivative of the inner function must be compensated by its reciprocal outside.

一个常见错误是在逆向链式法则中忘记除以x的系数。很多学生会错误地把∫ (3x + 2)⁴ dx写成(3x+

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