Integration Methods: Integrating Common Functions | 积分法：常见函数的积分计算

📚 Integration Methods: Integrating Common Functions | 积分法：常见函数的积分计算

Integration is one of the two central operations in calculus, alongside differentiation. In this article, we will explore how to integrate the most common functions you will encounter in A-Level mathematics: powers, exponentials, trigonometric functions, and combinations of these.

积分是微积分中与微分并列的两大核心运算之一。本文将探讨 A-Level 数学中常见函数的积分方法，包括幂函数、指数函数、三角函数及其组合。

1. The Basic Idea: Integration as Reverse Differentiation | 基本思想：积分是微分的逆运算

An indefinite integral, also called an antiderivative, is a function whose derivative equals the original function. If F'(x) = f(x), then we write ∫ f(x) dx = F(x) + C.

不定积分，也叫原函数，是一个导数等于原函数的函数。若 F'(x) = f(x)，则写作 ∫ f(x) dx = F(x) + C。

The constant C is essential because differentiating any constant gives zero. Therefore, adding any constant to F(x) does not change its derivative.

常数 C 必不可少，因为任何常数微分为零。因此，给 F(x) 加上任意常数不会改变其导数。

2. The Power Rule for xⁿ | 幂函数 xⁿ 的积分

For any constant power n, except n = -1, the integral of xⁿ is:

对于任何常数幂 n，除 n = -1 外，xⁿ 的积分为：

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C (n ≠ -1)

To check, differentiate xⁿ⁺¹/(n+1). By the power rule for derivatives, you get exactly xⁿ.

检验方法：对 xⁿ⁺¹/(n+1) 求导，根据幂函数微分法则，恰好得到 xⁿ。

For example, ∫ x³ dx = x⁴/4 + C. For a negative power such as x⁻², we have ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C.

例如，∫ x³ dx = x⁴/4 + C。对于负幂次如 x⁻²，有 ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C。

3. The Special Case: ∫ 1/x dx = ln|x| + C | 特殊情况：∫ 1/x dx = ln|x| + C

The power rule fails when n = -1 because it would require division by zero. Instead, the integral of x⁻¹ is the natural logarithm of the absolute value of x.

当 n = -1 时，幂函数积分法失效，因为会出现除以零。x⁻¹ 的积分是 x 绝对值的自然对数。

∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C

The absolute value is necessary because ln x is only defined for positive x, but 1/x is defined for all x ≠ 0. The derivative of ln|x| is 1/x for all x ≠ 0.

绝对值是必要的，因为 ln x 只对正数 x 有定义，而 1/x 对所有 x ≠ 0 都有定义。对 ln|x| 求导，对所有 x ≠ 0 都得到 1/x。

This result is used frequently in integration by substitution and in solving differential equations.

这一结果在换元积分法和解微分方程中频繁使用。

4. Exponential Functions | 指数函数

The natural exponential function eˣ is unique because it is its own derivative and also its own integral.

自然指数函数 eˣ 很独特，因为它既是自身的导数，也是自身的积分。

∫ eˣ dx = eˣ + C

For a general exponential base a > 0, a ≠ 1, the integral is:

对于一般底数 a > 0，a ≠ 1 的指数函数，积分为：

∫ aˣ dx = aˣ / ln a + C

For example, ∫ 2ˣ dx = 2ˣ / ln 2 + C. This follows because the derivative of aˣ is aˣ ln a.

例如，∫ 2ˣ dx = 2ˣ / ln 2 + C。这是由 aˣ 的导数为 aˣ ln a 得出的。

5. Trigonometric Functions | 三角函数

The integrals of the six basic trigonometric functions are standard results. The three most common are:

六个基本三角函数的积分是标准结果。最常用的三个是：

∫ sin x dx = -cos x + C

∫ cos x dx = sin x + C

∫ sec² x dx = tan x + C

Notice the minus sign in the integral of sin x. This is because the derivative of cos x is -sin x, so the derivative of -cos x is sin x.

注意 sin x 的积分中有负号。因为 cos x 的导数是 -sin x，所以 -cos x 的导数是 sin x。

Other standard results include:

其他标准结果包括：

∫ cosec² x dx = -cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = -cosec x + C
∫ tan x dx = -ln|cos x| + C = ln|sec x| + C

These can be verified by differentiating the right-hand side.

上述结果都可以通过对右边求导来验证。

6. Constant Multiples | 常数倍

If a function is multiplied by a constant, the integral is the constant multiplied by the integral of the function:

若函数乘以常数，其积分等于该常数乘以函数的积分：

∫ k f(x) dx = k ∫ f(x) dx

For example, ∫ 3x² dx = 3 ∫ x² dx = 3 * (x³/3) + C = x³ + C.

例如，∫ 3x² dx = 3 ∫ x² dx = 3 * (x³/3) + C = x³ + C。

This rule allows us to integrate expressions term by term after factoring out constants.

这条规则允许我们先提出常数，再逐项积分。

7. Sums and Differences: Linearity | 加减法与线性性

Integration is linear: the integral of a sum or difference is the sum or difference of the integrals.

积分具有线性性：和或差的积分等于积分的和或差。

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

For example, ∫ (x³ + cos x) dx = x⁴/4 + sin x + C. Remember to add a single constant C at the end.

例如，∫ (x³ + cos x) dx = x⁴/4 + sin x + C。记住最后只加一个常数 C。

The linearity of integration follows directly from the linearity of differentiation.

积分的线性性直接来源于微分的线性性。

8. Integrals of Linear Arguments | 线性复合函数的积分

For a linear expression ax + b, integration formulas extend naturally. If we know ∫ f(x) dx = F(x) + C, then:

对于线性表达式 ax + b，积分公式可以自然推广。若已知 ∫ f(x) dx = F(x) + C，则：

∫ f(ax + b) dx = (1/a) F(ax + b) + C

This is a special case of integration by substitution, sometimes called the reverse chain rule. For example:

这是换元积分法的特例，有时称为“反链式法则”。例如：

∫ (2x + 1)⁵ dx = (2x + 1)⁶ / (6 * 2) + C = (2x + 1)⁶ / 12 + C
∫ e³ˣ⁺² dx = (1/3)e³ˣ⁺² + C
∫ cos(4x – 1) dx = (1/4) sin(4x – 1) + C
∫ 1/(5x + 3) dx = (1/5) ln|5x + 3| + C

Always divide by the coefficient of x, which is a.

务必除以 x 的系数 a。

9. Definite Integrals and Area | 定积分与面积

A definite integral has upper and lower limits. It is evaluated using the antiderivative:

定积分有上下限，通过原函数来求值：

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a)

where F is an antiderivative of f, and a and b are the limits of integration.

其中 F 是 f 的一个原函数，a 和 b 是积分限。

For example, ∫₁³ x² dx = [x³/3]₁³ = (27/3) – (1/3) = 26/3.

例如，∫₁³ x² dx = [x³/3]₁³ = 27/3 – 1/3 = 26/3。

Definite integrals calculate the signed area under the curve y = f(x) between x = a and x = b. Area above the x-axis is positive, and area below the x-axis is negative.

定积分计算曲线 y = f(x) 在 x = a 与 x = b 之间的“有向面积”。x 轴上方的面积为正，下方的面积为负。

10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Many students make avoidable mistakes when integrating. Here are the most common ones:

许多学生在积分时会出现可避免的错误。以下是最常见的几种：

Mistake	Correction
Forgetting the constant + C	Always add + C for indefinite integrals.
Using the power rule for n = -1	Use ∫ x⁻¹ dx = ln\|x\| + C.
Forgetting the minus sign in ∫ sin x dx	Remember ∫ sin x dx = -cos x + C.
Forgetting to divide by a for linear arguments	∫ f(ax + b) dx = (1/a)F(ax + b) + C.

Always check your answer by differentiating it. If your result is correct, differentiation will return the original function.

每次都要通过求导来检查答案。如果结果正确，求导后会回到原函数。

11. Worked Examples | 综合例题

Let us apply the rules to a few complete examples.

让我们用完整的例子来应用上述规则。

Example 1: Find ∫ (3x² – 2eˣ + 5) dx.

例 1：求 ∫ (3x² – 2eˣ + 5) dx。

Solution: ∫ 3x² dx = x³; ∫ -2eˣ dx = -2eˣ; ∫ 5 dx = 5x. Therefore the answer is x³ – 2eˣ + 5x + C.

解答：∫ 3x² dx = x³；∫ -2eˣ dx = -2eˣ；∫ 5 dx = 5x。因此答案为 x³ – 2eˣ + 5x + C。

Example 2: Find ∫₀^π 2 sin x dx.

例 2：求 ∫₀^π 2 sin x dx。

Solution: ∫ 2 sin x dx = -2 cos x. Evaluate from 0 to π: (-2 cos π) – (-2 cos 0) = (-2 * -1) – (-2 * 1) = 2 – (-2) = 4.

解答：∫ 2 sin x dx = -2 cos x。从 0 到 π 代入：(-2 cos π) – (-2 cos 0) = (-2 * -1) – (-2 * 1) = 2 – (-2) = 4。

Example 3: Find ∫ (4/(2x – 1)) dx.

例 3：求 ∫ (4/(2x – 1)) dx。

Solution: Use the linear rule for 1/(2x – 1): ∫ 1/(2x – 1) dx = (1/2) ln|2x – 1| + C. Multiply by 4 to get 2 ln|2x – 1| + C.

解答：对 1/(2x – 1) 使用线性规则：∫ 1/(2x – 1) dx = (1/2) ln|2x – 1| + C。乘以 4 得到 2 ln|2x – 1| + C。

These examples show how combining the basic rules allows you to integrate a wide range of common functions.

这些例子显示，结合基本规则可以积分很多常见函数。

12. Conclusion | 总结

Mastering the integration of common functions is essential for A-Level mathematics. The key is to recognise each function type and apply the corresponding rule, then verify by differentiation.

掌握常见函数的积分对 A-Level 数学至关重要。关键在于识别函数类型、应用相应规则，并通过求导验证。

Practice makes perfect. Build a table of standard integrals, memorise the basic formulas, and remember to include the constant of integration for indefinite integrals.

熟能生巧。建议整理一张标准积分表，记住基本公式，并注意不定积分要加上积分常数。

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Integration Methods: Integrating Common Functions | 积分法：常见函数的积分计算