Exercise 21G.2: Definite Integral Evaluation | 练习21G.2：定积分计算

📚 Exercise 21G.2: Definite Integral Evaluation | 练习21G.2：定积分计算

Definite integrals form the backbone of many IB Mathematics problems, including those in Exercise 21G.2. Mastering the evaluation of definite integrals, especially via substitution, is essential for success in calculus and its applications. This article unpacks the core techniques, common pitfalls, and step-by-step strategies you need to confidently tackle any definite integral question.

定积分是许多 IB 数学题目的核心，包括练习 21G.2 中的内容。掌握定积分的计算，尤其是换元积分法，对于学好微积分及其应用至关重要。本文将剖析核心技巧、常见陷阱以及逐步解题策略，帮助你从容应对所有定积分题目。

1. Understanding Definite Integrals | 理解定积分

A definite integral represents the signed area between a curve and the x-axis over a closed interval [a, b]. The notation is ∫ₐᵇ f(x) dx, where a is the lower limit and b is the upper limit. The value is a number, not a function, and it is found using the Fundamental Theorem of Calculus.

定积分表示曲线与 x 轴在闭区间 [a, b] 上的有向面积。记法为 ∫ₐᵇ f(x) dx，其中 a 是积分下限，b 是积分上限。定积分的值是一个数字，而非函数，可通过微积分基本定理求得。

2. The Substitution Method (u-Substitution) | 换元积分法（u 代换）

The substitution method simplifies integrals by transforming a complicated integrand into a basic form. For definite integrals, you can either change the limits to match the new variable or substitute back to x after integration. Exercise 21G.2 often requires you to recognise the correct substitution.

换元法通过将复杂被积函数转化为基本形式来简化积分。对于定积分，你可以将积分限也转换为新变量的界限，或先换元积分再代回原变量。练习 21G.2 经常要求你选择合适的代换。

Let u = g(x) → du = g'(x) dx → ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du

设 u = g(x) → du = g'(x) dx → ∫ₐᵇ f(g(x))g'(x) dx = ∫_{g(a)}^{g(b)} f(u) du

3. Adjusting Limits of Integration | 调整积分限

When you perform u-substitution on a definite integral, you must replace the limits a and b with the corresponding u-values: u = g(a) and u = g(b). This keeps the evaluation consistent and avoids the need to substitute back to x. Many students forget this step, leading to errors.

在对定积分进行 u 代换时，必须将上下限 a 和 b 替换为相应的 u 值：u = g(a) 和 u = g(b)。这样做能保持计算结果一致，并省去代回原变量的麻烦。很多学生会忘记这一步，导致错误。

Step	Action	Example for u = x²
1	Write du in terms of dx	du = 2x dx
2	Find new limits for u	Lower: x=1 → u=1; Upper: x=3 → u=9
3	Rewrite integral entirely in u	∫₁³ (1/2) eᵘ du

4. Trigonometric Substitutions | 三角换元

Integrals involving √(a² – x²), √(a² + x²), or √(x² – a²) call for trigonometric substitutions. In Exercise 21G.2 you may see forms like √(1 – x²), where substituting x = sin θ simplifies the radical. Remember to convert the integration limits accordingly.

含有 √(a² – x²)、√(a² + x²) 或 √(x² – a²) 的积分需要用三角换元。在练习 21G.2 中可能遇到 √(1 – x²) 等形式，此时令 x = sin θ 可消除根号。切记同时转换积分限。

For ∫₀¹ √(1 – x²) dx, let x = sin θ → dx = cos θ dθ, limits: x=0 → θ=0; x=1 → θ=π/2.

对于 ∫₀¹ √(1 – x²) dx，令 x = sin θ → dx = cos θ dθ，积分限：x=0 → θ=0；x=1 → θ=π/2。

5. Integration of Exponential Functions | 指数函数的积分

Exponential integrands frequently appear with linear or quadratic exponents. If the integrand is of the form e^{kx}, integration is straightforward. However, when the exponent contains a function of x, such as e^{x²}, a u-substitution like u = x² is often required. Look for the derivative of the exponent as a factor.

指数型被积函数经常以一次或二次指数形式出现。如果形如 e^{kx}，积分很简单。但如果指数包含 x 的函数，例如 e^{x²}，往往需要 u = x² 这样的代换。要留意被积函数中是否恰好有指数部分的导数作为因子。

6. Using Symmetry to Simplify | 利用对称性化简

Definite integrals over symmetric intervals (e.g., [-a, a]) can often be simplified by checking parity. If f(x) is odd, ∫₋ₐᵃ f(x) dx = 0. If f(x) is even, ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx. This technique can save time and reduce algebraic errors in Exam-style problems related to Exercise 21G.2.

在对称区间（如 [-a, a]）上的定积分常可利用奇偶性简化。若 f(x) 为奇函数，则 ∫₋ₐᵃ f(x) dx = 0。若 f(x) 为偶函数，则 ∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx。运用这一技巧能节省时间、减少代数错误，非常适合练习 21G.2 这类考题。

7. Handling Improper Integrals (If Relevant) | 处理反常积分（如涉及）

Some questions extend to improper integrals with infinite limits or integrand discontinuities. In these cases, replace the problematic bound with a variable and take the limit. Although Exercise 21G.2 mainly covers proper integrals, being aware of this extension adds depth to your understanding.

有些题目会延伸到无穷区间或被积函数存在间断点的反常积分。此类情况下，用变量代替问题界限并取极限。虽然练习 21G.2 主要考察正常积分，但了解这一延伸能加深理解。

8. Common Errors in Definite Integrals | 定积分中的常见错误

• Forgetting to change limits when substituting.
• Misapplying the Fundamental Theorem: F(b) – F(a), not F(a) – F(b).
• Losing the differential dx during manipulation.
• Incorrectly handling negative signs when limits are reversed.

常见错误包括：换元时忘记改变积分限；错误应用基本定理（应为 F(b) – F(a) 而非 F(a) – F(b)）；代数变换中丢失微分 dx；积分限颠倒时符号处理不当。

9. Worked Examples Modeled on Exercise 21G.2 | 仿照练习21G.2的典型例题

Example 1: Evaluate ∫₀² 2x (x² + 1)³ dx.
Let u = x² + 1 → du = 2x dx, limits: x=0 → u=1, x=2 → u=5.
Integral becomes ∫₁⁵ u³ du = [u⁴/4]₁⁵ = (625/4) – (1/4) = 156.

示例1：计算 ∫₀² 2x (x² + 1)³ dx。
令 u = x² + 1 → du = 2x dx，积分限：x=0 → u=1，x=2 → u=5。
积分化为 ∫₁⁵ u³ du = [u⁴/4]₁⁵ = (625/4) – (1/4) = 156。

Example 2: Evaluate ∫₀^{π/2} sin³ x cos x dx.
Let u = sin x → du = cos x dx, limits: x=0 → u=0, x=π/2 → u=1.
Integral becomes ∫₀¹ u³ du = 1/4.

示例2：计算 ∫₀^{π/2} sin³ x cos x dx。
令 u = sin x → du = cos x dx，积分限：x=0 → u=0，x=π/2 → u=1。
积分化为 ∫₀¹ u³ du = 1/4。

10. Linking Substitution to Area and Applications | 将换元与面积及应用联系起来

Definite integrals are frequently used to find areas between curves, volumes of revolution, and total displacement in kinematics. In all these contexts, the ability to cleanly evaluate the integral with substitution is non-negotiable. Exercise 21G.2 builds the fluency needed for such applied problems.

定积分常用于计算曲线间的面积、旋转体体积以及运动学中的总位移。在所有这些情境下，流畅使用换元法进行定积分计算是必备能力。练习 21G.2 为这些应用题奠定了坚实的运算基础。

11. Checking Your Answers with Technology | 用技术工具检查答案

While IB expects students to perform integrations by hand, using your GDC to verify a definite integral’s numerical value is a smart exam strategy. After completing a problem from Exercise 21G.2, enter the integral into your calculator to confirm the result. This builds confidence and catches arithmetic slips.

虽然 IB 要求学生手动积分，但使用图形计算器验证定积分数值是一个聪明的应试策略。做完练习 21G.2 的题目后，将积分输入计算器进行核对，既能增强信心，也能发现计算失误。

12. Practice Mindset and Next Steps | 练习心态与后续步骤

Consistent practice of substitutions in definite integrals will make the process automatic. After mastering Exercise 21G.2, challenge yourself with mixed exercises that combine trigonometric identities, partial fractions, and integration by parts. This layered approach mirrors the diversity of IB exam questions.

坚持练习定积分的换元法会使解题过程自动化。掌握练习 21G.2 后，可以挑战混合使用三角恒等式、部分分式法和分部积分法的综合题目。这种分层训练与 IB 考试的多样性完美契合。

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