A-Level数学积分技巧：掌握∫(ax+b)ⁿdx与指数函数积分 | Integration of (ax+b)

Integration of functions of the form (ax+b)ⁿ 是 A-Level 数学纯数模块中的核心技能。看似简单——”把幂次加 1，除以新幂次和 x 系数”——但考试中频繁以变形形式出现，是许多考生的失分重灾区。本文系统梳理 (ax+b) 类型积分的五大变体，配套练习题解析，助你彻底攻克这一考点。

Integrating functions of the form (ax+b)ⁿ is a cornerstone skill in A-Level Pure Mathematics. The rule seems simple — “add 1 to the power, divide by the new power and the coefficient of x” — but exam questions routinely disguise it, making this a common pitfall. This article systematically covers five key variants of (ax+b) integration with worked examples to help you master the topic.

📐 核心公式 | The Core Formula

对于 n ≠ −1 的情况：
∫(ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + C
这就是”反向链式法则”（reverse chain rule）的直接应用。关键点：不仅要除以新幂次 (n+1)，还要除以内部函数 ax+b 的导数 a。忘记除 a 是最常见的错误。

For n ≠ −1:
∫(ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + C
This is a direct application of the reverse chain rule. The critical point: not only must you divide by the new power (n+1), you must also divide by a, the derivative of the inner function ax+b. Forgetting to divide by a is the single most common mistake.

🔢 知识点一：基本幂函数积分 | Basic Power Integration

例 1：∫(2x + 5)³ dx
解：n = 3, a = 2 → ∫(2x+5)³ dx = (2x+5)⁴ / (2 × 4) = (2x+5)⁴ / 8 + C

Example 1: ∫(2x + 5)³ dx
Solution: n = 3, a = 2 → ∫(2x+5)³ dx = (2x+5)⁴ / (2 × 4) = (2x+5)⁴ / 8 + C

例 2：∫(8 − 5x)⁴ dx
注意：这里 a = −5（不是 5！）→ ∫(8−5x)⁴ dx = (8−5x)⁵ / (−5 × 5) = −(8−5x)⁵ / 25 + C

Example 2: ∫(8 − 5x)⁴ dx
Watch out: here a = −5 (not 5!) → ∫(8−5x)⁴ dx = (8−5x)⁵ / (−5 × 5) = −(8−5x)⁵ / 25 + C

📏 知识点二：分母形式的积分 | Integrating Fractions with Linear Denominators

这是 n = −1 的特殊情况——公式不再适用！当被积函数为 1/(ax+b) 时：
∫ 1/(ax+b) dx = (1/a) · ln|ax+b| + C

This is the n = −1 special case — the power formula breaks! For integrands of the form 1/(ax+b):
∫ 1/(ax+b) dx = (1/a) · ln|ax+b| + C

例 3：∫ 1/(4x−3) dx = (1/4) · ln|4x−3| + C
例 4：∫ 3/(2x+1) dx = (3/2) · ln|2x+1| + C （常数因子提出后再积分）

Example 3: ∫ 1/(4x−3) dx = (1/4) · ln|4x−3| + C
Example 4: ∫ 3/(2x+1) dx = (3/2) · ln|2x+1| + C (factor out the constant, then integrate)

⚡ 知识点三：指数函数积分 | Integrating Exponential Functions with Linear Exponents

指数函数 e^(ax+b) 的积分也遵循反向链式法则：
∫ e^(ax+b) dx = (1/a) · e^(ax+b) + C

Integrating e^(ax+b) also follows the reverse chain rule:
∫ e^(ax+b) dx = (1/a) · e^(ax+b) + C

例 5：∫ e^(2x−3) dx = (1/2) e^(2x−3) + C
例 6：∫ 5e^(7−3t) dt = 5 × (−1/3) e^(7−3t) = −(5/3) e^(7−3t) + C

Example 5: ∫ e^(2x−3) dx = (1/2) e^(2x−3) + C
Example 6: ∫ 5e^(7−3t) dt = 5 × (−1/3) e^(7−3t) = −(5/3) e^(7−3t) + C

📐 知识点四：定积分应用 — 求曲线下方面积 | Definite Integrals — Area Under a Curve

定积分的核心步骤：先求不定积分 → 代入上下限 → 相减。关键陷阱：当 a 为负数且幂次为偶数时，符号处理需格外小心。

Core steps for definite integrals: find the indefinite integral → substitute bounds → subtract. Key trap: when a is negative and the power is even, sign handling requires extra care.

例 7：计算 ∫₀¹ (3x+1)² dx
解：F(x) = (3x+1)³ / (3×3) = (3x+1)³ / 9
F(1) = 64/9, F(0) = 1/9 → 结果 = 63/9 = 7

Example 7: Evaluate ∫₀¹ (3x+1)² dx
Solution: F(x) = (3x+1)³ / 9, F(1) = 64/9, F(0) = 1/9 → Result = 7

例 8：求 y = 12/(2x+1)³ 在 x=0 到 x=1 之间与坐标轴围成的面积
y = 12(2x+1)⁻³ → ∫ 12(2x+1)⁻³ dx = 12 × (2x+1)⁻² / (−2×2) = −3(2x+1)⁻² + C
面积 = [−3/(2x+1)²]₀¹ = −3/9 − (−3/1) = −1/3 + 3 = 8/3

Example 8: Area bounded by y = 12/(2x+1)³, x=0, x=1, and axes.
y = 12(2x+1)⁻³ → ∫ 12(2x+1)⁻³ dx = 12 × (2x+1)⁻² / (−2×2) = −3(2x+1)⁻² + C
Area = [−3/(2x+1)²]₀¹ = −3/9 − (−3/1) = 8/3

🧪 知识点五：已知导数求原函数 | Finding f(x) from f'(x)

当题目给出 f'(x) 和曲线上的一个点时，先积分得到含常数 C 的 f(x)，再代入已知点求 C。这是考试中最容易拿分也最容易丢分的题型——积分正确但忘记解 C，至少扣 2 分。

When given f'(x) and a point on the curve: first integrate to get f(x) with unknown constant C, then substitute the point to find C. This is simultaneously the easiest-to-score and easiest-to-lose-marks question type — correct integration followed by forgetting to solve for C costs at least 2 marks.

例 9：f'(x) = 8(2x−3)³, 曲线过点 (2, 6), 求 f(x)
f(x) = ∫ 8(2x−3)³ dx = 8 × (2x−3)⁴ / (2×4) = (2x−3)⁴ + C
代入 (2, 6)：6 = (4−3)⁴ + C → C = 5 → f(x) = (2x−3)⁴ + 5

Example 9: f'(x) = 8(2x−3)³, curve passes through (2, 6). Find f(x).
f(x) = ∫ 8(2x−3)³ dx = (2x−3)⁴ + C, then 6 = 1⁴ + C → C = 5 → f(x) = (2x−3)⁴ + 5

💡 学习建议 | Study Tips

• 牢记 n = −1 的特殊情况：当幂次为 −1 时必须切换到 ln 公式，尤其注意 1/(ax+b) 类型
• 养成”检查 a 的符号”的习惯：被积函数含减号（如 8−5x）时，a 为负数，积分结果会出现负号
• 定积分先求不定积分再代值：不要在不定积分阶段省略 +C，虽然定积分中 C 会抵消，但中间步骤写清楚可避免符号错乱
• 画图辅助面积题：曲线是否过 x 轴？是否需要分段积分？画一张粗略草图能减少 80% 的符号错误
• 计时练习 Solomon Press 习题：这份教材的题目覆盖了所有变体形式，每天做 10 道，两周即可形成肌肉记忆

• Memorise the n = −1 exception: when power is −1, switch to the ln formula — especially 1/(ax+b) variants
• Make “check the sign of a” a reflex: when the integrand contains a subtraction (e.g., 8−5x), a is negative and the integral will have a minus sign
• Write the full +C in indefinite integrals: even though C cancels in definite integrals, writing it in intermediate steps prevents sign confusion
• Sketch the curve for area problems: does the curve cross the x-axis? Do you need piecewise integration? A rough sketch eliminates 80% of sign errors
• Timed practice with Solomon Press worksheets: these cover all integration variants — 10 problems a day builds muscle memory in two weeks

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