📚 A-Level Maths: Integration of Power Functions x^n | A-Level数学：幂函数x^n的积分公式
In A-Level Mathematics, integration is a key skill that unlocks the calculation of areas, volumes, and many physical quantities. Among all integration rules, the power rule for functions of the form x^n is the most frequently used. This article explains the rule, its limitations, and how to apply it confidently in exams.
在 A-Level 数学中，积分是一项关键技能，它让我们能够计算面积、体积以及许多物理量。在所有的积分法则中，适用于形如 x^n 函数的幂法则最为常用。本文详细讲解该法则、其限制条件，以及如何在考试中自信地运用它。
1. The General Power Rule for Integration | 一般幂函数的积分法则
For any real constant n ≠ -1, the indefinite integral of x raised to the power n is found by increasing the exponent by 1 and dividing by the new exponent.
对任意实数 n ≠ -1，x 的 n 次幂的不定积分可以通过将指数加 1，再除以新指数来求得。
∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ -1)
Here, C is the constant of integration. This formula works for all real values of n except n = -1, because for n = -1 the denominator n+1 becomes zero.
这里的 C 是积分常数。该公式适用于除 n = -1 之外的所有实数 n，因为当 n = -1 时，分母 n+1 变为零。
2. Why Does This Formula Work? | 为什么这个公式成立？
The power rule for integration is the reverse of the power rule for differentiation. If we differentiate xⁿ⁺¹/(n+1), we use the chain rule in its simplest form:
幂函数积分法则正是幂函数微分法则的逆运算。如果对 xⁿ⁺¹/(n+1) 求导，使用最基础的导数法则，得到：
d/dx [ xⁿ⁺¹/(n+1) ] = (n+1)xⁿ/(n+1) = xⁿ
Therefore, the antiderivative of xⁿ is indeed xⁿ⁺¹/(n+1), provided n+1 ≠ 0.
因此，xⁿ 的原函数正是 xⁿ⁺¹/(n+1)，前提是 n+1 ≠ 0。
3. The Special Case n = -1 | 特殊情形 n = -1
When n = -1, we are integrating x⁻¹, which is the same as 1/x. The general rule cannot be used because it would involve division by zero.
当 n = -1 时，我们积分的是 x⁻¹，即 1/x。一般法则不能使用，因为它会导致除以零。
∫ x⁻¹ dx = ∫ 1/x dx = ln|x| + C
The absolute value is essential because the natural logarithm is only defined for positive inputs, while 1/x is defined for all x ≠ 0. In fact, the derivative of ln|x| is 1/x for both positive and negative x.
绝对值必不可少，因为自然对数仅对正数有定义，而 1/x 对所有 x ≠ 0 都有定义。事实上，ln|x| 的导数在 x 为正和负时都等于 1/x。
4. Worked Example: Integrating a Polynomial | 例题：多项式的积分
Let us evaluate the indefinite integral ∫ (6x² – 4x + 3) dx. We integrate each term separately.
让我们计算不定积分 ∫ (6x² – 4x + 3) dx。我们分别对每一项积分。
∫ (6x² – 4x + 3) dx = 6∫x² dx – 4∫x dx + 3∫x⁰ dx
Applying the power rule to each term:
对每一项使用幂法则：
6 · x³/3 – 4 · x²/2 + 3 · x + C = 2x³ – 2x² + 3x + C
Remember that the constant 3 is really 3x⁰, so its integral is 3x.
请记住，常数 3 实际上是 3x⁰，因此它的积分是 3x。
5. Worked Example: Negative and Fractional Powers | 例题：负指数与分数指数
The power rule also works for negative and fractional exponents, as long as n ≠ -1. For example, consider ∫ √x dx = ∫ x^(1/2) dx.
幂法则同样适用于负指数和分数指数，只要 n ≠ -1。例如，∫ √x dx = ∫ x^(1/2) dx。
∫ x^(1/2) dx = x^(3/2) / (3/2) + C = (2/3)x^(3/2) + C
Next, let us integrate a negative power: ∫ x⁻² dx.
接下来，我们积分一个负指数：∫ x⁻² dx。
∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C
Similarly, ∫ 1/x³ dx = ∫ x⁻³ dx = x⁻²/(-2) + C = -1/(2x²) + C.
类似地，∫ 1/x³ dx = ∫ x⁻³ dx = x⁻²/(-2) + C = -1/(2x²) + C。
6. Definite Integrals and Area under a Curve | 定积分与曲线下面积
To evaluate a definite integral from a to b, we first find the antiderivative F(x), then compute F(b) – F(a).
要计算从 a 到 b 的定积分，我们先求出原函数 F(x)，再计算 F(b) – F(a)。
∫ₐᵇ xⁿ dx = F(b) – F(a) = (bⁿ⁺¹ – aⁿ⁺¹)/(n+1) (n ≠ -1)
For example, ∫₀¹ 3x² dx = [x³]₀¹ = 1³ – 0³ = 1. This result represents the exact area between the curve y = 3x², the x-axis, and the lines x = 0 and x = 1.
例如，∫₀¹ 3x² dx = [x³]₀¹ = 1³ – 0³ = 1。这个结果表示曲线 y = 3x²、x 轴以及直线 x = 0 和 x = 1 之间的精确面积。
For n = -1, the definite integral becomes ∫ₐᵇ 1/x dx = ln|b| – ln|a| = ln|b/a|.
当 n = -1 时，定积分变为 ∫ₐᵇ 1/x dx = ln|b| – ln|a| = ln|b/a|。
7. Common Mistakes and Exam Tips | 常见错误与考试提示
Below are some common pitfalls when integrating power functions, along with tips to avoid them.
下面是积分幂函数时常见的错误陷阱，以及避免这些错误的提示。
- Forgetting +C: In indefinite integrals, always include the constant of integration. 忘记 +C：在不定积分中，务必加上积分常数。
- Using n = -1 with the general formula: Division by zero is undefined. Use ln|x| instead. 在 n = -1 时使用一般公式：除以零没有意义，应改用 ln|x|。
- Forgetting to simplify constants: Always cancel coefficients such as 6/3 or 4/2 before writing the final answer. 忘记化简常数：在写出最终答案前，一定要约分，例如 6/3 或 4/2。
- Sign errors with negative exponents: When n+1 is negative, the result is negative; be careful with minus signs. 负指数的符号错误：当 n+1 为负数时，结果会带负号，要小心处理符号。
- Omitting absolute value in ln: For ∫ 1/x dx, always write ln|x|, not just ln x. 在 ln 中遗漏绝对值：对于 ∫ 1/x dx，应写 ln|x|，而不是仅写 ln x。
8. Practice Problems | 练习题
Try the following questions on your own, then compare with the answers below.
请先独立完成以下题目，再与下面的答案对照。
| Question / 题目 | Answer / 答案 |
| ∫ (4x³ – 2x + 1) dx | x⁴ – x² + x + C |
| ∫ 1/x² dx | -1/x + C |
| ∫ √x dx | 更多咨询请联系16621398022（同微信）
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