📚 IB Mathematics: Systematic Methods and Techniques for Integration | IB数学：积分的系统化方法与技巧梳理
Integration is one of the central operations in IB Mathematics, appearing in both Analysis and Approaches and Applications and Interpretation. Mastering it requires more than memorising formulas: you need a clear strategy for choosing the right technique from a given function.
积分是IB数学的核心运算之一，在“分析与方法”和“应用与解释”两门课程中都会出现。掌握积分不只是背公式，更需要形成清晰的策略，能够针对不同函数选择合适的方法。
1. Understanding Integration as Inverse Differentiation | 理解积分作为微分的逆运算
onstant of integration. The Fundamental Theorem of Calculus links this to the definite integral: the signed area under f(x) from a to b is F(b) − F(a).
积分是微分的逆运算。若 F′(x) = f(x)，则 F(x) 是 f(x) 的一个原函数，且 ∫ f(x) dx = F(x) + C，其中 C 为积分常数。微积分基本定理将这一关系与定积分联系起来：f(x) 从 a 到 b 的有向面积等于 F(b) − F(a)。
For IB questions, you should always check whether a result can be obtained by reversing a known derivative. This saves time and reduces errors, especially in multiple-choice or short-response questions.
在IB题目中，你应该经常检查一个结果是否可以通过逆转已知导数得到。这样既能节省时间，也能减少错误，尤其是在选择题和简答题中。
2. Standard Integrals You Must Know | 必须掌握的标准积分公式
The base table of integrals is non-negotiable. These forms appear repeatedly and are the foundation of every substitution and manipulation.
基础积分表必须熟练掌握。这些形式会反复出现，也是所有换元和变形的基础。
| Function | Integral |
| k (constant) | kx + C |
| xⁿ | xⁿ⁺¹/(n+1) + C, n ≠ −1 |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec²x | tan x + C |
| 1/(1+x²) | arctan x + C |
| 1/√1−x² | arcsin x + C |
You should also know the linearity rule: ∫ [a·f(x) + b·g(x)] dx = a∫ f(x) dx + b∫ g(x) dx.
你还应该掌握线性法则：∫ [a·f(x) + b·g(x)] dx = a∫ f(x) dx + b∫ g(x) dx。
3. The Power Rule and Algebraic Manipulation | 幂法则与代数变形
Many IB integrals look complicated but become simple after algebraic rewriting. Radicals, reciprocals, and products can often be changed into powers of x.
许多IB积分看起来复杂，但经过代数改写后会变得非常简洁。根式、倒数和乘积通常可以改写为 x 的幂。
- √x = x¹ᐟ², so ∫ √x dx = x³ᐟ² / (3/2) + C = (2/3)x√x + C.
- 1/x² = x⁻², so ∫ x⁻² dx = x⁻¹ / (−1) + C = −1/x + C.
- Expand products before integrating: ∫ x(x+1) dx = ∫ (x²+x) dx = x³/3 + x²/2 + C.
applying the power rule. For example, (x³+2)/x = x² + 2/x, so ∫ (x³+2)/x dx = x³/3 + 2 ln|x| + C.
在运用幂法则之前，一定要先对分式逐项化简。例如 (x³+2)/x = x² + 2/x，因此 ∫ (x³+2)/x dx = x³/3 + 2 ln|x| + C。
4. Integration by Substitution | 换元积分法
see a composite function multiplied by the derivative of its inner function, let u = g(x).
换元法是链式法则的逆用。当你看到一个复合函数乘以其内层函数的导数时，可以令 u = g(x)。
∫ f(g(x))·g′(x) dx = ∫ f(u) du, where u = g(x)
Let u = x², so du = 2x dx. Then the integral becomes ∫ cos u du = sin u + C = sin(x²) + C.
例如：计算 ∫ 2x·cos(x²) dx。令 u = x²，则 du = 2x dx。于是积分变为 ∫ cos u du = sin u + C = sin(x²) + C。
For definite integrals, you can either convert the limits to u-values or substitute back before evaluating using x-limits. In IB exams, converting limits is usually cleaner and avoids unnecessary simplification.
对定积分而言，你可以把上下限转换为对应的 u 值，也可以先换回 x 再用原上下限计算。在IB考试中，转换上下限通常更简洁，也能避免不必要的化简。
5. Integration by Parts | 分部积分法
Integration by parts is used for products of unrelated function types, such as x·eˣ or x·ln x. The formula comes from the product rule.
分部积分法适用于不同类型函数相乘的情况，例如 x·eˣ 或 x·ln x。该公式来源于乘积法则。
∫ u dv = uv − ∫ v du
To choose u, many IB students use the LIATE rule: choose u in the order of Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. This order generally makes the remaining integral easier.
选择 u 时，许多IB学生会使用 LIATE 规则：按对数函数、反三角函数、代数函数、三角函数、指数函数的顺序选取 u。这个顺序通常能让剩下的积分更简单。
Example: ∫ x·eˣ dx. Let u = x, so du = dx; let dv = eˣ dx, so v = eˣ. Then ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C.
例如：∫ x·eˣ dx。令 u = x，则 du = dx；令 dv = eˣ dx，则 v = eˣ。于是 ∫ x·eˣ dx = x·eˣ − ∫ eˣ dx = x·eˣ − eˣ + C。
6. Definite Integrals and Their Properties | 定积分及其性质
umulation of a quantity between two limits. The notation ∫[a,b] f(x) dx means F(b) − F(a).
定积分表示某个量在两个边界之间的累积。记号 ∫[a,b] f(x) dx 表示 F(b) − F(a)。
Knowing properties of definite integrals can simplify problems dramatically:
掌握定积分的性质可以大大简化问题：
- ∫[a,b] f(x) dx = −∫[b,a] f(x) dx.
- ∫[a,c] f(x) dx = ∫[a,b] f(x) dx + ∫[b,c] f(x) dx.
- If f is odd, then ∫[−a,a] f(x) dx = 0.
- If f is even, then ∫[−a,a] f(x) dx = 2∫[0,a] f(x) dx.
These properties are particularly useful for symmetric functions and for avoiding difficult antiderivatives in exam questions.
这些性质在处理对称函数和避免复杂原函数时尤其有用。
7. Area Under Curves and Between Curves | 曲线下面积与曲线间面积
The area between a curve and the x-axis is given by ∫[a,b] |f(x)| x. The absolute value is essential because the definite integral itself gives signed area.
曲线与 x 轴之间的面积由 ∫[a,b] |f(x)| dx 给出。绝对值至关重要，因为定积分本身表示的是有向面积。
he area between two curves y = f(x) and y = g(x), where f(x) ≥ g(x) on [a,b], the area is:
对于两条曲线 y = f(x) 与 y = g(x) 之间的面积，若在 [a,b] 上 f(x) ≥ g(x)，则面积为：
Area = ∫[a,b] [f(x) − g(x)] dx
If the curves intersect, you must split the integral at each intersection point and use the correct upper-lower ordering in each interval. Otherwise you may calculate negative area.
如果两条曲线相交，你必须在每个交点处拆分积分，并在每个区间内正确判断上下位置关系，否则可能计算出负面积。
8. Volumes of Revolution | 旋转体体积
region under y = f(x) is rotated around the x-axis, the volume of the resulting solid is:
当 y = f(x) 下方的区域绕 x 轴旋转时，所得旋转体的体积为：
V = π ∫[a,b] [f(x)]² dx
When rotating around the y-axis, express x as a function of y and integrate with respect to y:
当绕 y 轴旋转时，将 x 表示为 y 的函数，并对 y 积分：
V = π ∫[c,d] [x(y)]² dy
In IB, always state the formula and show the square clearly. Missing the π or squaring only part of the expression are common careless errors.
在IB考试中，一定要写出体积公式并清楚地展示平方。漏写 π 或只对表达式的一部分进行平方都是常见的粗心错误。
9. Trigonometric Integrals and Identities | 三角函数的积分与恒等式
Squares and products of sine and cosine often require identities before integration. The double-angle identities are your best tool.
正弦和余弦的平方或乘积通常需要先使用恒等式再积分。二倍角公式是最实用的工具。
sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2
Therefore:
因此：
- ∫ sin²x dx = x/2 − sin(2x)/4 + C
- ∫ cos²x dx = x/2 + sin(2x)/4 + C
- ∫ tan x dx = ln|sec x| + C
For products like sin 3x·cos 2x, use product-to-sum identities, or try a substitution if the derivative factor is present.
对于像 sin 3x·cos 2x 这样的乘积，可以使用积化和差公式；如果存在导数因子，也可以尝试换元。
10. Partial Fractions for Rational Functions | 有理函数的部分分式
When integrating a rational function P(x)/Q(x), first check whether the numerator has lower degree than the denominator. If not, perform polynomial long division first.
在积分有理函数 P(x)/Q(x) 时，首先要检查分子的次数是否低于分母。如果不是，需要先进行多项式长除法。
Then decompose the fraction into simpler components. For distinct linear factors, write:
然后将分式分解为更简单的部分。对于互不相同的线性因式，写成：
Q(x) = (ax+b)(cx+d) → P/Q = A/(ax+b) + B/(cx+d)
Example: ∫ 2/(x²−1) dx. Since 2/((x−1)(x+1)) = 1/(x−1) − 1/(x+1), the integral is ln|x−1| − ln|x+1| + C = ln|(x−1)/(x+1)| + C.
例如：∫ 2/(x²−1) dx。由于 2/((x−1)(x+1)) = 1/(x−1) − 1/(x+1)，所以积分为 ln|x−1| − ln|x+1| + C = ln|(x−1)/(x+1)| + C。
For repeated factors or quadratic factors, use the corresponding partial fraction template given in the formula booklet or learned in class.
对于重因式或二次因式，请使用公式手册或课堂上给出的相应部分分式模板。
11. Improper Integrals and Limits | 反常积分与极限
Improper integrals appear when an interval is infinite or when the function has a vertical asymptote inside the interval. They are evaluated using limits.
反常积分出现在积分区间无限或函数在区间内存在垂直渐近线的情形。它们需要通过极限来计算。
∫[a,∞] f(x) dx = lim(t→∞) ∫[a,t] f(x) dx
,∞] 1/x² dx = lim(t→∞) [−1/t + 1] = 1. The integral converges because the limit is finite.
例如：∫[1,∞] 1/x² dx = lim(t→∞) [−1/t + 1] = 1。由于极限是有限的，所以该反常积分收敛。
If the limit does not exist, the integral diverges. Always state whether the improper integral converges or diverges in your final answer.
如果极限不存在，则积分发散。在最终答案中，一定要说明反常积分是收敛还是发散。
12. Systematic Strategy and Common Pitfalls | 系统化步骤与常见陷阱
Use the following decision sequence for any indefinite integral:
在计算任何不定积分时，可以按以下决策顺序进行：
- Step 1: Simplify algebraically: expand, divide, or rewrite roots as powers. 步骤一：代数化简：展开、做除法或把根号写成幂。
- Step 2: Ask whether the integrand matches a standard form. 步骤二：判断被积函数是否符合标准积分形式。
- Step 3: Look for a composite function and its derivative; if present, use substitution. 步骤三：寻找复合函数及其导数；若存在，使用换元法。
- Step 4: For products, choose u by LIATE and use integration by parts. 步骤四：对于乘积，用 LIATE 选择 u 并使用分部积分法。
- Step 5: For rational functions, use partial fractions after checking the degree. 步骤五：对于有理函数，检查次数后使用部分分式。
- Step 6: Differentiate your answer to check. 步骤六：对答案求导进行验证。
Common mistakes include forgetting +C, dropping absolute values in ln|x|, not changing limits in definite substitution, sign errors in integration by parts, and confusing area with signed integral. In the IB exam, clear working and final simplification are rewarded, so keep your layout systematic and check every term.
常见错误包括忘记常数 C、在 ln|x| 中漏掉绝对值、定积分换元时不更换上下限、分部积分时出现符号错误，以及混淆面积与有向积分。在IB考试中，清晰的步骤和最终化简会得分，因此请保持系统化的书写格式，并逐项检查。
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