Integration Techniques for Edexcel A-Level Mathematics | Edexcel A-Level 数学积分方法

📚 Integration Techniques for Edexcel A-Level Mathematics | Edexcel A-Level 数学积分方法

Integration is a central topic in Edexcel A-Level Mathematics. You need to recognise standard forms, apply substitution and parts correctly, and use integration to solve area problems. This article brings together the core techniques and exam tips you should master before the written paper.

积分是 Edexcel A-Level 数学的核心主题。你需要识别标准形式，正确使用换元法和分部积分法，并用积分解决面积问题。本文汇总你在笔试前必须掌握的核心技巧和考试要点。

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1. Standard Integrals and the Constant of Integration | 标准积分与积分常数

The indefinite integral of a function f(x) is written as ∫ f(x) dx and represents the general antiderivative F(x) + C, where F'(x) = f(x). The constant C is essential because differentiation removes any constant term.

函数 f(x) 的不定积分写作 ∫ f(x) dx，表示一般原函数 F(x) + C，其中 F'(x) = f(x)。常数 C 必不可少，因为微分会消去任何常数项。

∫ f(x) dx = F(x) + C, F'(x) = f(x)

For example, ∫ 2x dx = x² + C because d/dx(x²) = 2x. Without C, you lose a whole family of antiderivatives.

例如，∫ 2x dx = x² + C，因为 d/dx(x²) = 2x。没有 C，你会丢失一整族原函数。

Function	Integral
xⁿ, n ≠ -1	xⁿ⁺¹/(n+1) + C
1/x	ln|x| + C
eˣ	eˣ + C
eᵃˣ	(1/a)eᵃˣ + C
sin x	-cos x + C
cos x	sin x + C

These standard results form the building blocks for more complicated integrals. You should be able to quote them quickly under exam pressure.

这些标准结果是构建更复杂积分的基础。你应该能够在考试压力下快速写出它们。

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2. Reversing Differentiation: The Fundamental Concept | 微分的逆运算：基本概念

Integration can often be checked by differentiation. If you think a result is F(x), then differentiating F(x) should return the original integrand.

积分通常可以通过微分来检验。如果你认为结果是 F(x)，那么对 F(x) 求导应该得到原来的被积函数。

This reverse relationship is the reason why every differentiation rule has an integration counterpart. The chain rule leads to substitution, the product rule leads to integration by parts, and the quotient rule often links to partial fractions.

这种逆关系是每个微分规则都有对应积分规则的原因。链式法则对应换元积分，乘积法则对应分部积分，商法则通常联系到部分分式。

For example, since d/dx(sin 3x) = 3 cos 3x, it follows that ∫ 3 cos 3x dx = sin 3x + C. Dividing both sides by 3 gives ∫ cos 3x dx = (1/3) sin 3x + C.

例如，因为 d/dx(sin 3x) = 3 cos 3x，所以 ∫ 3 cos 3x dx = sin 3x + C。两边除以 3 得到 ∫ cos 3x dx = (1/3) sin 3x + C。

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3. Integrating Powers and Polynomials | 幂函数与多项式的积分

For any real power n ≠ -1, the power rule for integration states that ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C.

对于任意实数幂 n ≠ -1，幂函数积分法则是 ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ -1

Polynomials are integrated term by term. For example, ∫ (3x² – 4x + 5) dx = x³ – 2x² + 5x + C.

多项式逐项积分。例如，∫ (3x² – 4x + 5) dx = x³ – 2x² + 5x + C。

Remember that constants are treated as x⁰. The special case n = -1 gives the natural logarithm: ∫ x⁻¹ dx = ln|x| + C.

请记住常数可以看作 x⁰。特殊情形 n = -1 给出自然对数：∫ x⁻¹ dx = ln|x| + C。

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4. Integrating Exponentials and 1/x | 指数函数与 1/x 的积分

The exponential function eˣ is its own derivative and its own antiderivative, so ∫ eˣ dx = eˣ + C. For a constant multiple in the exponent, ∫ eᵃˣ dx = (1/a)eᵃˣ + C.

指数函数 eˣ 的导数与反导数都是它本身，因此 ∫ eˣ dx = eˣ + C。对于指数中的常数倍，∫ eᵃˣ dx = (1/a)eᵃˣ + C。

∫ eᵃˣ dx = (1/a)eᵃˣ + C

The reciprocal function integrates to the natural logarithm of the absolute value: ∫ (1/x) dx = ln|x| + C. This is used in many rational integrals.

倒数函数的积分是绝对值的自然对数：∫ (1/x) dx = ln|x| + C。这在许多有理积分中都会用到。

For example, ∫ (4/x + 2e³ˣ) dx = 4 ln|x| + (2/3)e³ˣ + C. Notice how each term is integrated separately and the constants are combined into one C.

例如，∫ (4/x + 2e³ˣ) dx = 4 ln|x| + (2/3)e³ˣ + C。注意每一项分别积分，常数合并成一个 C。

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5. Integrating Trigonometric Functions | 三角函数的积分

The basic trigonometric integrals are ∫ sin x dx = -cos x + C and ∫ cos x dx = sin x + C.

基本三角积分是 ∫ sin x dx = -cos x + C 和 ∫ cos x dx = sin x + C。

∫ sin x dx = -cos x + C, ∫ cos x dx = sin x + C

For linear angles, ∫ sin(ax+b) dx = -(1/a)cos(ax+b) + C and ∫ cos(ax+b) dx = (1/a)sin(ax+b) + C.

对于线性角，∫ sin(ax+b) dx = -(1/a)cos(ax+b) + C，∫ cos(ax+b) dx = (1/a)sin(ax+b) + C。

The integral of sec² x is tan x + C, and the integral of tan x is ln|sec x| + C, though the latter is often quoted via substitution.

sec² x 的积分是 tan x + C，tan x 的积分是 ln|sec x| + C，不过后者通常通过换元得到。

Be careful with the sign when integrating sine: the derivative of cos x is -sin x, so the integral of sin x must include a negative sign.

积分正弦时要小心符号：cos x 的导数是 -sin x，所以 sin x 的积分必须带一个负号。

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6. Integration by Substitution | 换元积分法

Substitution is the reverse of the chain rule. For an integral of the form ∫ f(g(x))g'(x) dx, set u = g(x), so du = g'(x) dx, and transform the integral into ∫ f(u) du.

换元法是链式法则的逆运算。对于 ∫ f(g(x))g'(x) dx 形式的积分，设 u = g(x)，则 du = g'(x) dx，把积分化为 ∫ f(u) du。

∫ f(g(x))g'(x) dx = ∫ f(u) du, u = g(x)

For example, to find ∫ 2x(x²+1)³ dx, let u = x²+1, then du = 2x dx. The integral becomes ∫ u³ du = u⁴/4 + C = (x²+1)⁴/4 + C.

例如，求 ∫ 2x(x²+1)³ dx，设 u = x²+1，则 du = 2x dx。积分变为 ∫ u³ du = u⁴/4 + C = (x²+1)⁴/4 + C。

For definite integrals, either change the limits to u-values or substitute back to x before evaluating. If you change limits, you do not need to return to the original variable.

对于定积分，要么把上下限换成 u 值，要么在求值前代回 x。如果替换了上下限，就不需要再回到

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