A-Level Maths: Integration Techniques Complete Guide | A-Level 数学：积分技巧完全指南

Introduction

Integration is one of the two central pillars of calculus, alongside differentiation. While differentiation follows a fairly mechanical set of rules, integration often requires creativity, pattern recognition, and a toolkit of techniques. For A-Level Mathematics students, mastering integration is essential not only for the pure mathematics papers but also for mechanics and statistics applications.

Integration (积分) is the inverse operation of differentiation. While differentiation asks “what is the rate of change?”, integration asks “given the rate of change, what is the original quantity?”. This fundamental duality makes integration indispensable across all branches of mathematics and physics.

This article provides a comprehensive guide to the integration techniques required for A-Level Mathematics, covering standard integrals, integration by substitution, integration by parts, partial fractions, parametric integration, and differential equations. Each section includes worked examples in both English and Chinese.

1. Standard Integrals – The Foundation

Before diving into advanced techniques, you must have the following standard integrals memorised. These are the building blocks that every integration problem reduces to:

在学习进阶技巧之前，你必须熟记以下标准积分：

Function f(x)	Integral
x^n (n not equal to -1)	x^(n+1)/(n+1) + C
1/x	ln|x| + C
e^x	e^x + C
sin x	-cos x + C
cos x	sin x + C
sec^2 x	tan x + C
1/(1+x^2)	arctan x + C
1/sqrt(1-x^2)	arcsin x + C

Key Exam Tip: Always add the constant of integration +C. In A-Level mark schemes, omitting it typically costs one mark per indefinite integral. 永远记得加上积分常数 +C。

2. Integration by Substitution

Substitution is arguably the most powerful and frequently tested integration technique. The idea is to replace a complicated expression with a simpler variable, transforming the integral into one of the standard forms above.

换元法是最常考的积分技巧。核心思想是用一个更简单的变量替换复杂表达式。

The Method Steps

1. Choose a substitution u = g(x) that simplifies part of the integrand.
2. Differentiate to find du/dx = g'(x), so dx = du / g'(x).
3. Rewrite the entire integral in terms of u only.
4. Integrate with respect to u.
5. Substitute back to x (for indefinite integrals) or evaluate new limits (for definite integrals).

1. 选择替换 u = g(x) 简化被积函数。2. 求导 du/dx = g'(x)。3. 将积分仅用 u 重写。4. 对 u 进行积分。5. 代回 x。

Example 1

Evaluate: Integral of 2x(x^2 + 1)^4 dx

Solution: Let u = x^2 + 1. Then du/dx = 2x, so dx = du/(2x). Substituting: Integral of 2x * u^4 * du/(2x) = Integral of u^4 du = u^5/5 + C = (x^2 + 1)^5/5 + C.

Example 2 – Trigonometric Substitution: Evaluate integral of sqrt(1 – x^2) dx from x = 0 to x = 1. Let x = sin(theta). Then dx = cos(theta) d(theta). Limits: when x=0, theta=0; when x=1, theta=pi/2. Integral becomes integral_0^(pi/2) cos^2(theta) d(theta) = pi/4.

3. Integration by Parts

Integration by parts is the integral counterpart of the product rule for differentiation. The formula is: Integral of u(dv/dx)dx = uv – Integral of v(du/dx)dx.

分部积分法是微分乘法法则的积分对应。公式：Integral of u(dv/dx)dx = uv – Integral of v(du/dx)dx。

The LIATE Rule

Use the LIATE rule to choose u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick u based on what appears first in LIATE.

LIATE 法则选择 u：L 对数、I 反三角、A 代数、T 三角、E 指数。

Example 3

Evaluate: Integral of x * e^x dx. By LIATE, let u = x (algebraic), dv/dx = e^x. Then du/dx = 1, v = e^x. Result: e^x(x – 1) + C.

Example 4 – The ln x trick: Integral of ln x dx = x*ln x – x + C. (Write as integral of 1 * ln x dx, let u = ln x).

4. Partial Fractions

When the integrand is a rational function where the denominator factorises, partial fractions break it into simpler fractions that integrate to logarithms or arctangents.

当被积函数是有理函数且分母可因式分解时，部分分式将其分解为简单分式之和。

• Distinct linear factors: 1/[(x-a)(x-b)] = A/(x-a) + B/(x-b)
• Repeated linear factors: 1/[(x-a)^2] = A/(x-a) + B/(x-a)^2

Example 5

Evaluate: Integral of (3x+5)/[(x-1)(x+2)] dx. Decompose: = 8/[3(x-1)] + 1/[3(x+2)]. Integral = (8/3)ln|x-1| + (1/3)ln|x+2| + C.

5. Parametric Integration

When a curve is defined parametrically as x = f(t), y = g(t), the area under the curve is: Area = Integral of y * (dx/dt) dt.

Example: x = t^2, y = t^3 – t, t from 0 to 1. dx/dt = 2t. Area = Integral_0^1 (t^3 – t)*2t dt = 4/15.

6. Separable Differential Equations

For equations of the form dy/dx = f(x)*g(y), separate variables and integrate: Integral of (1/g(y)) dy = Integral of f(x) dx.

Example: dy/dx = 2xy, y(0) = 3. Separate: (1/y)dy = 2x dx. Integrate: ln|y| = x^2 + C. Thus y = 3e^(x^2).

7. Trapezium Rule

For numerical integration: Integral from a to b of f(x)dx is approximately h/2 * [y_0 + y_n + 2(y_1 + … + y_(n-1))] where h = (b-a)/n.

梯形法则：h/2 * [y_0 + y_n + 2(y_1 + … + y_(n-1))]。Overestimates when concave up, underestimates when concave down。

8. Common Pitfalls and Exam Tips

• Don’t confuse integral of e^(x^2) dx with a standard integral – it has no elementary antiderivative.
• Check your answer by differentiating – the fastest verification method.
• For definite integrals, always convert the limits when using substitution.
• Remember the modulus signs in integral of 1/x dx = ln|x| + C.
• Factor constants out early to simplify.
• Practice trigonometric identities – sin^2 + cos^2 = 1, double-angle formulas.

不要把 e^(x^2) 误认为标准积分。通过求导检查答案。定积分换元时转换积分限。记住 ln|x| 的绝对值。尽早提出常数因子。练习三角恒等式。

Conclusion

Integration is a skill that improves dramatically with practice. The techniques covered here – standard integrals, substitution, parts, partial fractions, parametric integration, and separable differential equations – form the complete toolkit for A-Level Mathematics. Work through past paper questions systematically, starting with single-technique problems before tackling mixed exercises. Remember: the hardest part of integration is often recognising which technique to apply, not executing the technique itself.

积分是一个通过练习可以显著提高的技能。本文涵盖的标准积分、换元法、分部积分法、部分分式、参数积分和可分离微分方程构成了 A-Level 数学的完整工具箱。

Good luck with your studies! 祝学习顺利!