📚 Integrating xn | 积分 xn
Integration is one of the two fundamental operations in calculus, often introduced as the reverse process of differentiation. For functions of the form xn, where n is a real number, there is a simple and powerful formula that allows us to find the antiderivative quickly, provided that n is not equal to -1. This article guides you through the rule, its special cases, and its many applications in the Edexcel A-Level Mathematics syllabus. Understanding how to integrate xn correctly lays the foundation for tackling more complex problems involving polynomials, rational powers, definite integrals, and even kinematics.
积分是微积分中的两大基本运算之一，常被视作微分的逆过程。对于形如 xn 的函数（n 为实数），有一个简单且强大的公式可以快速求出原函数，前提是 n 不等于 −1。本文带你掌握这一积分法则、它的特殊情况以及在 Edexcel A-Level 数学大纲中的诸多应用。正确积分 xn 是攻克莱多项式、有理次幂、定积分乃至运动学问题的基础。
1. The Power Rule for Integration | 幂函数积分法则
If f(x) = xn, where n ≠ -1, then the indefinite integral is given by ∫ xn dx = xn+1 / (n+1) + C. This is the power rule for integration. To see why it works, differentiate xn+1/(n+1) with respect to x: the (n+1) factor comes down, cancels the denominator, and leaves xn. Thus integration simply reverses this process. The rule holds for any real exponent n other than -1, including positive integers, negative integers, fractions, and irrational numbers.
如果 f(x) = xn 且 n ≠ −1，则不定积分公式为 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。这就是幂函数积分法则。为了理解它的来源，对 xⁿ⁺¹/(n+1) 关于 x 求导：指数 (n+1) 下移，与分母中的 (n+1) 约掉，得到 xⁿ。因此积分恰好是这个微分过程的逆。该法则对不等于 −1 的任何实数指数 n 都成立，包括正整数、负整数、分数甚至无理数。
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1
2. Understanding the Constant of Integration | 理解积分常数
C. The reason is that differentiation strips away any added constant: d/dx (F(x) + C) = F' o account for all possible vertical translations of the function. For example, ∫ 2x dx = x2 upward by 3 units. In applications, an initial condition is used to determine C uniquely.
每个不定积分都必须加上积分常数，记作 C。原因在于微分会使任何加上的常数消失：d/dx (F(x) + C) = F'(x) + 0 = f(x)。因此，在反求原函数时，我们必须考虑所有可能的纵向平移。例如，∫ 2x dx = x² + C。取 C = 0 得到经过原点的抛物线，而 C = 3 则将它向上平移 3 个单位。在具体应用中，会借助初始条件唯一确定 C 的值。
3. Integrating Polynomials Term by Term | 逐项积分多项式
f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx, and constants can be factored out. For instance, consider ∫ (3x² + 2x − 5) dx. Integrate term by term: ∫ 3x² dx = 3 · x³/3 = x³, ∫ 2x dx = 2 · x²/2 = x², and ∫ (−5) dx = −5x. Adding the constant C gives the antiderivative x³ + x² − 5x + C. Always remember to increase the power by 1 and divide by the new power.
利用积分的线性性质，我们可以对多项式的每一项分别使用幂法则进行积分。加法法则表明 ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx，而常数因子可以提出。例如，计算 ∫ (3x² + 2x − 5) dx：逐项积分，∫ 3x² dx = 3 · x³/3 = x³，∫ 2x dx = 2 · x²/2 = x²，∫ (−5) dx = −5x。加上积分常数 C 后得到原函数 x³ + x² − 5x + C。务必记住：指数加 1，再除以新的指数。
- For a term a xⁿ, the integral is (a/(n+1)) xⁿ⁺¹, provided n ≠ -1.
- 对于形如 a xⁿ 的项，积分结果为 (a/(n+1)) xⁿ⁺¹，前提是 n ≠ −1。
4. Special Case: n = -1 | 特殊情况：n = −1
es it clear that a different antiderivative is needed. In fact, ∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C, where ln|x| . This result is fundamental and often appears in Edexcel exam questions, especially when integrating rational functions or solving differential equations.
当 n = −1 时，幂法则将会出现除以零的情况，这是未定义的。公式本身的极限形式表明需要另一种原函数。实际上，∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C，其中 ln|x| 是 x 的绝对值的自然对数。加上绝对值可以保证函数在正数和负数区域都有定义（x = 0 除外）。这一结果是基础中的基础，频繁出现在 Edexcel 考题中，尤其在积分有理函数或解微分方程时。
∫ 1/x dx = ln|x| + C
5. Integrating with Negative and Fractional Exponents | 负指数和分数指数的积分
Many expressions involving roots and reciprocals can be rewritten as xn before integration. For example, the square root √x is the same as x1/2, and 1/x² is x-2. Once in power form, the standard rule applies. To integrate ∫ √x dx, write ∫ x½ dx = x3/2 / (3/2) = (2/3)x3/2 + C. For ∫ (1/x³) dx, rewrite as ∫ x-3 dx = x-2 ctional and negative powers confidently is essential for Edexcel Pure Mathematics 1 and 2.
许多含有根号和倒数的表达式都可以先改写成 xⁿ 的形式再进行积分。例如，平方根 √x 等同于 x½，而 1/x² 可写为 x⁻²。一旦化为幂函数形式，便可使用标准公式。计算 ∫ √x dx 时，写作 ∫ x½ dx = x3/2 / (3/2) = (2/3) x3/2 + C。对于 ∫ (1/x³) dx，则改写为 ∫ x⁻³ dx = x⁻²/(−2) = −1/(2x²) + C。熟练处理分数指数和负指数是应对 Edexcel 纯数 1、2 的关键。
- ∫ x½ dx = (2/3) x3/2 + C
- ∫ x−2 dx = −x−1 + C = −1/x + C
- ∫ x½ dx = (2/3) x3/2 + C
- ∫ x⁻² dx = −x⁻¹ + C = −1/x + C
6. Definite Integration of xn | xⁿ 的定积分
A definite integral ∫_ab s the net area between the curve y = f(x) and the x-axis from x = a to x = b. For f(x) = xn, the procedure is to find the antiderivative F(x) = xn+1 ative is (1/3)x³. Substituting the limits gives (8/3) − (1/3) = 7/3. The constant C cancels out in definite integrals, so it is not written.
定积分 ∫ₐᵇ f(x) dx 计算的是曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的净面积。对于 f(x) = xⁿ，步骤是求出原函数 F(x) = xⁿ⁺¹/(n+1)，然后计算 F(b) − F(a)。这就是微积分基本定理的第二部分。例如，求 ∫₁² x² dx：原函数为 (1/3) x³。代入上下限得 (8/3) − (1/3) = 7/3。在定积分中，常数 C 会被抵消，因此书写时不再写出。
∫ₐᵇ xⁿ dx = [xⁿ⁺¹/(n+1)]ₐᵇ, n ≠ -1
7. Applying Integration to Kinematics | 积分在运动学中的应用
tion (a) are related by calculus. If acceleration is given as a function of time, a(t) = tn e constants that are determined by initial conditions, such as the velocity or displacement at t = 0. For example, if a(t) = 2t and v(0) = 3, then v(t) = ∫ 2t dt = t² + C. Using v(0) = 3 gives C = 3, so v(t) = t² + 3. Integrating again gives s(t) = (1/3)t³ + 3t + D, where D is found from initial displacement.
在 Edexcel 的力学部分，位移 (s)、速度 (v) 和加速度 (a) 通过微积分联系在一起。如果加速度以时间 t 的函数形式给出，例如 a(t) = tⁿ，则速度可以通过积分求得：v(t) = ∫ a(t) dt。再进行一次积分即可得到位移：s(t) = ∫ v(t) dt。两次积分都会引入常数，这些常数由初始条件（如 t = 0 时的速度或位移）确定。例如，若 a(t) = 2t 且 v(0) = 3，则 v(t) = ∫ 2t dt = t² + C。利用 v(0) = 3 得到 C = 3，因此 v(t) = t² + 3。再次积分得 s(t) = (1/3)t³ + 3t + D，D 由初始位移求出。
8. Finding the Equation of a Curve | 求曲线方程
nt C. Adding a point that lies on the curve provides the additional information needed to find C. For instance, suppose dy/dx = 3x² − 2 and the curve passes through (1, 4). Integrate to get y = x³ − 2x + C. Substitute x = 1 and y = 4: 4 = (1)³ − 2(1) + C ⇒ 4 = −1 + C ⇒ curve is y = x³ − 2x + 5. This technique is common in Edexcel Pure Mathematics questions.
当导数 dy/dx 以关于 x 的表达式给定时，通过积分可以还原出原曲线 y = f(x)，但会带有常数 C。若再知道曲线上一点，就可以求出 C。例如，设 dy/dx = 3x² − 2，且曲线经过点 (1, 4)。积分得 y = x³ − 2x + C。代入 x = 1, y = 4：4 = (1)³ − 2(1) + C ⇒ 4 = −1 + C ⇒ C = 5。因此曲线方程为 y = x³ − 2x + 5。此类题型在 Edexcel 纯数考试中非常常见。
9. Common Mistakes to Avoid | 常见错误避免
ation C in indefinite integrals; using the power rule when n = -1 (giving an impossible division by zero); miscalculating the new exponent, e.g. adding 1 incorrectly or mishandling negative signs; mixing up the order of subtraction in definite integrals, that is, writing F(a) − F(b) instead of F(b) − F(a); failing to simplify fractions, and omitting brackets when substituting limits. Double-checking by differentiating the result is an excellent way to catch these mistakes.
许多学生在积分题上因可避免的错误而失分。最常见的错误包括：不定积分中漏写积分常数 C；当 n = −1 时仍使用幂法则（导致除以零的非法操作）；新指数计算失误，如加 1 算错或负号处理不当；定积分代入时颠倒相减顺序，即写成了 F(a) − F(b) 而非 F(b) − F(a)；忘记约分，以及在代入上下限时漏掉括号。对结果进行求导检验是发现这些错误的好方法。
- Forgetting '+ C' loses a mark in indefinite integration.
- Applying ∫ x⁻¹ dx formula wrongly loses method marks.
- 漏写 '+ C' 在不定积分中会扣分。
- 错误套用 ∫ x⁻¹ dx 的公式会失去方法分。
10. Exam Tips for Edexcel A-Level | Edexcel A-Level 考试技巧
When tackling integration problems in Edexcel exams, always show clear, logical steps. Write down the integral, then the intermediate antiderivative with the new exponent, before evaluating limits. For definite integrals, use square brackets to indicate the antiderivative with limits: [F(x)]ab. If the question says 'hence' or 'using part (a)', remember to use the previously established result — do not start from scratch. Manage your time wisely and leave a few minutes to verify answers by differentiation, especially for high-mark questions. Keep an eye on the index; if you see 1/x, switch to the ln rule immediately.
在 Edexcel 考试中解答积分题时，始终要展示清晰、有逻辑的计算步骤。先写出积分式，再写出新指数下的原函数，最后代入上下限。对于定积分，使用方括号将带上下限的原函数括起来：[F(x)]ₐᵇ。如果题目中出现 "hence" 或 "using part (a)" 等提示，记得使用前面已得出的结论，不要从头再来。合理分配时间，留出几分钟通过求导检验答案，尤其是高分题。留神被积函数的指数；一见到 1/x，立刻切换为 ln 法则。
11. Practice Example with Step-by-Step Solution | 逐步解题示例
Let us work through a question that combines several aspects of integrating xn. Question: (a) Find ∫ (2x − 1/√x) dx. (b) Hence evaluate ∫₁⁴ (2x − 1/√x) dx. (c) A curve has gradient function 2x − 1/√x and passes through (4, 10). Find the equation of the curve.
我们来做一道综合了多种 xⁿ 积分技巧的题目。问题：(a) 求 ∫ (2x − 1/√x) dx。(b) 据此计算 ∫₁⁴ (2x − 1/√x) dx。(c) 一曲线的导数函数为 2x − 1/√x，且经过点 (4, 10)，求曲线方程。
Solution (a): Rewrite the integrand in power form. 2x stays as 2x¹, and 1/√x becomes x-1/2. So the integral is ∫ (2x − x-½) dx. Use the power rule: ∫ 2x dx = 2 · x²/2 = x²; ∫ x-½ dx = x½ / (½) = 2x½. Therefore the antiderivative is x² − 2x½ + C.
解 (a)：将被积函数改写成幂形式。2x 保持 2x¹，1/√x 变为 x⁻¹ᐟ²。所以积分式为 ∫ (2x − x⁻¹²) dx。应用幂法则：∫ 2x dx = 2 · x²/2 = x²；∫ x⁻¹² dx = x¹² / (1/2) = 2x¹²。因此原函数为 x² − 2x¹² + C。
ion (b): Evaluate the definite integral using the antiderivative (without C). ∫₁⁴ (2x − 1/√x) dx = [x² − 2√x]₁⁴. At x=4: 4² − 2√4 = 16 − 4 = 12. At x=1: 1² − 2√1 = 1 − 2 = −1. Subtract lower from upper: 12 − (−1) = 13.
解 (b)：利用原函数（去掉 C）计算定积分。∫₁⁴ (2x − 1/√x) dx = [x² − 2√x]₁⁴。在 x=4 处：4² − 2√4 = 16 − 4 = 12。在 x=1 处：1² − 2√1 = 1 − 2 = −1。上限减下限：12 − (−1) = 13。
Solution (c): The curve's equation is y = ∫ (2x − 1/√x) dx = x² − 2√x + C. Substitute (4, 10): 10 = 4² − 2√4 + C → 10 = 16 − 4 + C → 10 = 12 + C → C = −2. Therefore the curve is y = x² − 2√x − 2.
解 (c)：曲线方程为 y = ∫ (2x − 1/√x) dx = x² − 2√x + C。代入 (4, 10)：10 = 4² − 2√4 + C → 10 = 16 − 4 + C → 10 = 12 + C → C = −2。因此曲线为 y = x² − 2√x − 2。
12. Summary and Key Points | 摘要与要点
The integration of xn is a cornerstone of A-Level calculus. The formula ∫ xn dx = xn+1/(n+1) + C works for all n ≠ −1. The special case n = −1 gives ∫ 1/x dx = ln|x| ed area under a curve. In mechanics and coordinate geometry, integration recovers quantities from rates of change when initial conditions are supplied. Mastering these techniques and avoiding common pitfalls will build confidence and accuracy in Edexcel Mathematics examinations.
对 xⁿ 的积分是 A-Level 微积分的基石。公式 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C 适用于所有 n ≠ −1 的情形。特殊情况 n = −1 给出 ∫ 1/x dx = ln|x| + C。积分具有线性性质：可以逐项处理加法与常数倍乘。定积分利用原函数来计算曲线下方的带号面积。在力学和坐标几何中，当已知变化率并给出初始条件时，积分可以从变化率恢复出原量。掌握这些技巧并避开常见陷阱，会为你在 Edexcel 数学考试中增添信心与准确度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图