📚 Review Set 25C: Conquering Integration for IB Math | 复习集25C：攻克IB数学积分难题
In the IB Mathematics: Analysis and Approaches course (and also relevant for Applications and Interpretation), Review Set 25C offers a challenging collection of integration problems that tie together the core techniques you have studied. From reversing differentiation to computing areas under curves, solving differential equations, and handling improper integrals, this review set demands both fluency in symbolic manipulation and conceptual understanding. The following guide walks you through the essential skills and strategies, with every explanation paired in English and Chinese, so you can build confidence and precision before your exam.
在IB数学:分析与方法的课程中（同样适用于应用与解释），复习集25C提供了一组具有挑战性的积分题，融合了你所学的各种核心技巧。从反向求导到计算曲线下的面积，再到求解微分方程和处理反常积分，这套复习题既要求熟练的符号运算能力，也考验对概念的深层理解。下面的指南将带你梳理必备的技能和策略，每条解释均以中英双语对照呈现，帮助你在考试前建立信心和准确度。
1. Essential Integration Formulas | 基本积分公式
Before tackling the mixed problems in Review Set 25C, you must be able to recall standard antiderivatives instantly. The power rule, logarithmic form, exponential and trigonometric integrals are the building blocks.
在攻克复习集25C的混合题之前，你必须能够立即回忆起标准的反导数。幂函数法则、对数形式、指数函数积分以及三角函数的积分都是基础模块。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1
This is the reverse of the power rule for derivatives. Remember to increase the exponent by one and divide by the new exponent.
这是导数幂函数法则的逆运算。记住将指数加一，再除以新的指数。
∫ 1/x dx = ln|x| + C
The absolute value inside the natural logarithm is crucial because the domain of 1/x includes negative x-values.
自然对数里必须加绝对值，这是因为 1/x 的定义域包含负数值。
∫ eˣ dx = eˣ + C
Exponential functions with base e remain unchanged upon integration, apart from the constant of integration.
以 e 为底的指数函数积分后除了积分常数外保持不变。
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C
Notice the sign change when integrating sine; it is a common slip in exams.
注意正弦积分时符号的变化；这是考试中常犯的错误。
ax+b) + C, which are direct extensions. Fluency in these basics saves time and prevents careless mistakes in the later sections of the review set.
你也可能见到线性复合函数积分，例如 ∫ sin(ax+b) dx = −(1/a) cos(ax+b) + C，这些都是直接推广。熟练掌握这些基础可以节省时间，并避免在复习集后续部分出现粗心错误。
2. Integration by Substitution (U-Substitution) | 换元积分法
Many problems in Review Set 25C require recognising an 'inner function' whose derivative appears elsewhere in the integrand. Substitution transforms the integral into a simpler standard form.
复习集25C中的许多题目都需要识别出“内部函数”，且其导数出现在被积函数中的某处。通过换元可将积分转化为更简单的标准形式。
The typical pattern is to let u = g(x), so du = g' ing u = x² gives du = 2x dx, so the integral becomes ∫ cos u du = sin u + C = sin(x²) + C.
典型的模式是令 u = g(x)，则 du = g'(x) dx，然后将被积函数完全用 u 表示。例如，考虑 ∫ 2x cos(x²) dx。令 u = x² 得 du = 2x dx，于是积分变为 ∫ cos u du = sin u + C = sin(x²) + C。
rigonometric identities and exponential expressions, so be ready for nested forms like ∫ esin x cos x dx.
对于定积分，记得在求值之前将积分限换成 u 值。这样避免了原函数换回 x 的步骤。在IB考试中，换元法经常与三角恒等式和指数表达式结合出现，因此要做好应对嵌套形式的准备，例如 ∫ esin x cos x dx。
Sketch your substitution clearly; a well-organised solution shows the examiner that you understand the chain rule in reverse.
清晰地写出换元过程；一份条理清楚的解答可以向阅卷人展示你懂得逆向运用链式法则。
3. Integration by Parts | 分部积分法
When the integrand is a product of functions that are not related by a simple derivative, integration by parts is the tool of choice. The formula is derived from the product rule for differentiation:
当被积函数是两个函数的乘积且无法通过简单的导数关系处理时，分部积分法是首选工具。该公式源于微分的乘法法则：
∫ u dv = uv − ∫ v du
Selecting which part to differentiate (u) and which to integrate (dv) requires practice. The LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) helps prioritise u: pick u as the function appearing earlier in the list.
选择哪一部分求导（作为 u）、哪一部分积分（作为 dv）需要多加练习。LIATE 规则（对数、反三角、代数、三角、指数）有助于确定 u 的优先顺序：选取在列表中较早出现的函数作为 u。
For example, in ∫ x eˣ dx, let u = x (algebraic) and dv = eˣ dx. Then du = dx and v = eˣ, giving ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ arts or combinations with substitution, so always check if a simpler method applies first.
例如，在 ∫ x eˣ dx 中，设 u = x（代数函数），dv = eˣ dx。则 du = dx，v = eˣ，得到 ∫ x eˣ dx = x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。复习集25C经常包含需要多次分部积分或与换元法结合的积分题，因此务必先检查是否存在更简单的方法。
A common pitfall is forgetting the negative sign in the cycle for trigonometric-exponential products; carefully track your signs and constants.
一个常见的陷阱是在处理三角函数与指数函数的乘积时忘记了负号；细致地追踪符号和常数。
4. Definite Integrals and Area | 定积分与面积
In Review Set 25C, definite integrals are not just numerical evaluations—they represent the signed area between a curve and the x-axis. Understanding the geometric meaning helps you split the integral correctly when the function crosses the axis.
在复习集25C中，定积分不仅仅是数值计算——它表示曲线与 x 轴之间的有向面积。理解其几何意义有助于在函数穿过 x 轴时正确分割积分区间。
To find the total area enclosed, compute ∫ |f(x)| ed by them. Always sketch the region first, even if the question does not require a graph.
要计算总围成面积，可以通过确定 f(x) 为正或负的区间来计算 ∫ |f(x)| dx。或者，计算两条曲线之差 (f(x) − g(x)) 的积分，以求出它们所围成的面积。务必先画出区域草图，即便题目未要求作图。
The fundamental theorem of calculus links differentiation and integration: if F' echnology (GDC) to check your answers, but always show the analytical steps as required by the IB.
微积分基本定理将微分与积分联系起来：若 F' = f，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。可以利用图形计算器检查答案，但切记要按照IB的要求展示分析步骤。
5. Volumes of Revolution | 旋转体体积
A classic application in the problem set is calculating the volume of a solid generated by rotating a region about the x-axis or y-axis. The formula for rotation about the x-axis is:
这套习题中的一个经典应用是计算将区域绕 x 轴或 y 轴旋转所得的立体体积。绕 x 轴旋转的公式为：
V = π ∫ₐᵇ [f(x)]² dx
For regions between two curves, the volume is π ∫ ([outer radius]² − [inner radius]²) dx.
如果绕 y 轴旋转，需将 x 表示为 y 的函数 x = g(y)，并使用 V = π ∫ₐᵇ [g(y)]² dy。对于两条曲线之间的区域，体积为 π ∫ ([外半径]² − [内半径]²) dx。
around lines other than the axes, such as y = c. Adjust the radius by subtracting the line's coordinate: e.g., radius = |f(x) − c|. Always square the radius before integrating.
复习集25C可能包含绕坐标轴以外的直线（如 y = c）旋转所产生的体积。通过减去直线的坐标来调整半径：例如，半径 = |f(x) − c|。始终先将半径平方再进行积分。
Watch for units and exact forms (leaving π in the answer). A quick sanity check: compare your volume to a bounding cylinder's volume to see if it is plausible.
注意单位和精确形式（答案中保留 π）。快速检验：将所得体积与包围它的圆柱体体积进行比较，看看是否合理。
6. Integration Leading to Inverse Trig Functions | 得到反三角函数的积分
Standard forms that yield inverse trigonometric functions appear frequently in the review set. Recognising them avoids lengthy substitutions.
在复习集中，产生反三角函数的标准形式频繁出现。认出它们可以避免冗长的换元过程。
∫ 1/√a² − x² dx = arcsin(x/a) + C, ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C
ritten by completing the square to the form ∫ 1/((x+2)²+3²) dx, yielding an arctan result.
这些积分常常隐藏在配方法或因式分解之后。例如，∫ 1/(x²+4x+13) dx 可通过配方法改写为 ∫ 1/((x+2)²+3²) dx，从而得到一个 arctan 结果。
A common mistake is forgetting the constant multiplier 1/a in the arctan formula; always differentiate your answer mentally to verify.
一个常见错误是忘记 arctan 公式中的常系数 1/a；务必在心里求导以验证结果。
7. Solving Simple Differential Equations | 求解简单微分方程
). In Review Set 25C, you will see equations that model growth, decay, or rates of change.
积分是求解形如 dy/dx = f(x) g(y) 的可分离一阶微分方程的关键。在复习集25C中，你将见到用于模拟增长、衰减或变化率的方程。
n. IB marks often require you to express the final answer in the form y = f(x) explicitly.
分离变量：写成 (1/g(y)) dy = f(x) dx，然后对两边积分。如果给出了初始条件，则用其求出积分常数。IB评分常常要求将最终的答案以显式形式 y = f(x) 表达。
Pay attention to the domain of the solution. For instance, if ln|y| appears, the absolute value can be dropped only if the initial condition guarantees y > 0. This justification shows deeper understanding.
注意解的定义域。例如，如果出现 ln|y|，只有当初始条件保证 y > 0 时才能去掉绝对值。这样的解释能展现出更深的理解。
8. Integration by Partial Fractions | 部分分式积分法
Rational functions where the denominator factorises into distinct linear factors call for partial fractions. This technique is explicitly used in some of the set's more advanced problems.
当分母可分解为不同的一次因子时，有理函数需要用到部分分式法。该技巧在这套题目的一些较难问题中被明确使用。
Decompose a fraction like (2x+1)/(x(x+1)) into A/x + B/(x+1), solve for A and B, then integrate each term separately, typically yielding logarithmic expressions. Extend the method to repeated linear factors and irreducible quadratics carefully, using the correct form: for (ax²+bx+c) term, the numerator is linear.
将像 (2x+1)/(x(x+1)) 这样的分式分解为 A/x + B/(x+1)，解出 A 和 B，然后分别积分每一项，通常会得到对数表达式。谨慎地将该方法推广到重因子和不可约二次因式，并使用正确的形式：对于 (ax²+bx+c) 项，其分子是一次式。
Combining partial fractions with u-substitution or completing the square is a favourite in IBHL exam papers; be systematic in equating coefficients or substituting convenient x-values to find the constants.
将部分分式与 u 代换或配方法结合是IBHL考卷中的热门题目；系统地使用系数相等法或代入特殊 x 值来求出常数。
9. Improper Integrals and Convergence | 反常积分与收敛性
Review Set 25C may include integrals with infinite limits or integrands with vertical asymptotes within the integration interval. These improper integrals require limits to be evaluated.
复习集25C可能包含无限积分限或在积分区间内有垂直渐近线的被积函数。这些反常积分需要通过极限求值。
diverges. A classic example is ∫₁∞ 1/x² dx, which converges, whereas ∫₁∞ 1/x dx diverges.
对于从 a 到 ∞ 的积分 ∫ f(x) dx，计算 lim(b→∞) ∫ₐᵇ f(x) dx。如果极限存在且有限，积分收敛；否则发散。一个经典例子是 ∫₁∞ 1/x² dx 收敛，而 ∫₁∞ 1/x dx 发散。
Be careful with discontinuities: split the integral at the point of discontinuity and treat each side as a limit. In IB exams, you may be asked to determine the value if convergent, or simply to state whether it converges.
对于不连续点要格外小心：在不连续点处将积分拆分，并将每一侧都作为极限处理。在IB考试中，可能要求你在收敛时求出其值，或者仅仅陈述它是否收敛。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
Even well-prepared students lose marks on integration due to small, avoidable slips. Review Set 25C is designed to expose these weaknesses before the real exam.
即使准备充分的学生也会因微小但可避免的失误而在积分题上失分。复习集25C的目的正是在真正考试前暴露这些弱点。
Forgetting the +C in indefinite integrals: always write it at the end.
在不定积分中忘记写 +C：始终要在最后加上。
Misapplying the power rule when n = -1; remember it leads to ln, not division by zero.
错误地将幂函数法则用于 n = -1 的情况；记住此时要得到 ln，而不是除以零。
Inconsistent limits in u-substitution: change limits to u-values or revert to x correctly; never mix them.
在 u 代换中积分限不一致：将积分限换成 u 值，或者正确换回 x；绝不能混淆使用。
Sign errors when integrating trigonometric functions or when using integration by parts: double-check with differentiation.
在三角积分或分部积分时出现符号错误：通过求导双重检查。
Not checking the answer with technology (GDC) when permitted: use it as a verification tool.
在允许使用图形计算器时没有检查答案：把它当作验证工具来用。
Practise writing clear, logically flowing solutions. In IB, communication is assessed, so showing each algebraic step can earn method marks even if the final answer is slightly off.
练习书写清晰、逻辑流畅的解答。在IB中，表达过程也会被评估，因此即使最终答案略有偏差，展示每一个代数步骤都可能赢得方法分。
11. Applying Integration to Kinematics and Real-World Contexts | 积分在运动学及实际情境中的应用
s where velocity, acceleration, and displacement are linked by integration. Given a(t) = v'(t) = s''(t), you integrate to recover v(t) and s(t) using initial values.
IB试卷青睐情境问题，复习集25C也包含了通过积分将速度、加速度和位移联系起来的应用题。给定 a(t) = v'(t) = s''(t)，通过积分并利用初值求出 v(t) 和 s(t)。
so v(t)=t²+3. Further integration yields displacement. Always include units if specified.
例如，若物体以 a(t) = 2t 加速，且初速度 v(0)=3，则 v(t) = ∫ 2t dt = t² + C；代入 t=0 得到 C=3，因此 v(t)=t²+3。再进行积分即可得到位移。若有规定，始终要包含单位。
Other scenarios include rate of water flow, population growth, or marginal cost in economics. Identifying the correct initial condition and interpreting the constant of integration in context is a skill Review Set 25C helps develop.
其他场景包括水流速率、人口增长或经济学中的边际成本。识别正确的初始条件，并在情境中解释积分常数，这是复习集25C能够帮助培养的技巧。
12. Final Review Strategy and Self-Reflection | 终极复习策略与自我反思
Working through Review Set 25C is not just about finding answers—it's about diagnosing your own understanding. Mark each question: which technique was tested? Did you recognise it quickly? If you struggled, return to that topic's section in this guide and recreate the worked example without looking.
做完复习集25C并不仅仅是为了找到答案——更是为了诊断自身的理解程度。为每道题做标记：它考察了哪种技巧？你是否迅速识别出来？如果遇到困难，回到本指南中相应主题的章节，在不看的情况下重新做出例题。
Use a mistake journal: write down the exact error, why it happened, and a corrected version. This transforms careless slips into lasting learning. Then, attempt the hardest three problems again the next day. Spaced repetition is scientifically proven to strengthen long-term memory.
使用错题本：记录具体的错误、发生的原因以及更正后的版本。这能将粗心的失误转化为持久的学习。然后，在第二天再次尝试最难的三道题。间隔重复经科学证明可以加强长期记忆。
Finally, simulate an exam condition when redoing the set: time each problem, and force yourself to rely only on formula booklet allowed resources. Confidence comes from preparation, and Review Set 25C is one of your best preparation tools.
最后，在重新做题时模拟考试环境：为每道题计时，并强制自己只依靠允许的公式手册等资源。自信来源于准备，而复习集25C就是你最好的备考工具之一。
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