Integration: Basic Concepts and Common Methods | 积分基本概念与常见方法

📚 Integration: Basic Concepts and Common Methods | 积分基本概念与常见方法

Integration is one of the two core operations in calculus, alongside differentiation. It is used to find areas, volumes, accumulated quantities, and to reverse the process of differentiation.

积分是微积分中与微分并列的两大核心运算之一。它用于求面积、体积、累积量，以及逆转微分的过程。

1. Antiderivatives and Indefinite Integrals | 原函数与不定积分

An antiderivative of a function f(x) is a function F(x) such that F′(x) = f(x). The indefinite integral of f(x) with respect to x is written as ∫f(x) dx and represents the family of all antiderivatives of f(x).

函数 f(x) 的原函数是满足 F′(x) = f(x) 的函数 F(x)。f(x) 关于 x 的不定积分记作 ∫f(x) dx，它表示 f(x) 的所有原函数的集合。

Since the derivative of a constant is zero, any two antiderivatives differ by a constant. Therefore we write:

由于常数的导数为零，任意两个原函数之间相差一个常数。因此我们写作：

∫f(x) dx = F(x) + C

where C is an arbitrary constant called the constant of integration.

其中 C 为任意常数，称为积分常数。

2. The Power Rule for Integration | 积分幂法则

The power rule for integration is the reverse of the power rule for differentiation. For n ≠ −1,

积分幂法则是微分幂法则的逆运算。当 n ≠ −1 时，

∫xⁿ dx = xⁿ⁺¹⁄(n+1) + C

For n = −1, the rule does not apply; instead we have:

当 n = −1 时，该法则不适用，此时有：

∫x⁻¹ dx = ln|x| + C

Remember to include the absolute value inside the logarithm because the domain of ln(x) is x > 0, while x⁻¹ is defined for x ≠ 0.

注意对数内要加绝对值，因为 ln(x) 的定义域为 x > 0，而 x⁻¹ 在 x ≠ 0 时有定义。

3. Basic Integration Rules | 基本积分法则

Integration is linear, which means it can be applied term by term and constants can be pulled out:

积分具有线性性，这意味着可以逐项积分，常数可以提到积分号外：

∫[af(x) + bg(x)] dx = a∫f(x) dx + b∫g(x) dx

The integral of a sum is the sum of the integrals.

和的积分等于积分的和。
A constant factor can be moved outside the integral sign.

常数因子可以移到积分号外。
These rules hold for all linear combinations, not just two terms.

这些规则适用于任意线性组合，不仅限于两项。

4. Integrals of Common Functions | 常见函数的积分

You should be able to recall the integrals of basic functions quickly. The table below lists the most important ones for IB Mathematics.

你应该能够快速回忆起基本函数的积分。下表列出了IB数学中最重要的几个。

f(x)	∫f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹⁄(n+1) + C
1⁄x = x⁻¹	ln\|x\| + C
eˣ	eˣ + C
aˣ (a > 0, a ≠ 1)	aˣ⁄(ln a) + C
sin x	−cos x + C
cos x	sin x + C
sec² x	tan x + C
1⁄(1+x²)	arctan x + C
1⁄√(1−x²)	arcsin x + C

5. Definite Integrals and the Fundamental Theorem | 定积分与微积分基本定理

A definite integral has lower and upper limits of integration:

定积分具有积分的下限和上限：

∫ₐᵇ f(x) dx

The Fundamental Theorem of Calculus connects differentiation and integration. If F(x) is an antiderivative of f(x), then:

微积分基本定理将微分与积分联系起来。若 F(x) 是 f(x) 的一个原函数，则：

∫ₐᵇ f(x) dx = F(b) − F(a)

This theorem allows us to evaluate definite integrals exactly by finding an antiderivative and computing the difference at the endpoints.

该定理使我们能够通过求原函数并计算端点的差值来精确计算定积分。

6. Properties of Definite Integrals | 定积分的性质

Several properties are frequently used when manipulating definite integrals:

在处理定积分时，有几个性质经常使用：

∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx

交换上下限会改变积分的符号。
∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx for any c in [a,b].

对于 [a,b] 中的任意 c，积分可以分段相加。
∫ₐᵃ f(x) dx = 0

上下限相同时积分为零。
If f(x) ≥ 0 on [a,b], then ∫ₐᵇ f(x) dx ≥ 0.

若 f(x) 在 [a,b] 上非负，则定积分也非负。
∫ₐᵇ [f(x) + g(x)] dx = ∫ₐᵇ f(x) dx + ∫ₐᵇ g(x) dx

和的积分等于积分的和。

7. Integration by Substitution | 换元积分法

Integration by substitution is the reverse of the chain rule. It is used when the integrand contains a composite function multiplied by the derivative of the inner function.

换元积分法是链式法则的逆运算。当被积函数包含复合函数与内层函数导数的乘积时使用。

Steps for substitution:

换元法的步骤：

Choose u = g(x) where g(x) is the inner function.

选择 u = g(x)，其中 g(x) 是内层函数。
Compute du = g′(x) dx.

计算 du = g′(x) dx。
Rewrite the integral in terms of u and du.

将积分改写为关于 u 和 du 的形式。
Integrate with respect to u, then substitute back.

对 u 积分，然后再代回原变量。

Example: ∫2x cos(x²) dx. Let u = x², du = 2x dx. Then the integral becomes ∫cos u du = sin u + C = sin(x²) + C.

例如：∫2x cos(x²) dx。令 u = x²，du = 2x dx，则积分变为 ∫cos u du = sin u + C = sin(x²) + C。

8. Integration by Parts | 分部积分法

Integration by parts is the reverse of the product rule. It is used when the integrand is a product of two functions that are not related by a simple derivative.

分部积分法是乘积法则的逆运算。当被积函数是两个函数的乘积，且这两个函数之间没有简单的导数关系时使用。

∫u dv = uv − ∫v du

Choose u to be a function that becomes simpler when differentiated, and dv to be the remaining part that can be integrated easily.

选择 u 为微分后变简单的函数，dv 为剩余部分且容易积分。

A common acronym is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential — choose u in that order.

常用口诀“LIATE”：对数函数、反三角函数、代数函数、三角函数、指数函数——按此顺序选择 u。

9. Integration of Rational Functions | 有理函数的积分

A rational function is a ratio of two polynomials. Some rational functions can be integrated directly, while others require algebraic manipulation or partial fractions.

有理函数是两个多项式的比。有些有理函数可以直接积分，而其他的则需要代数变形或部分分式分解。

If the numerator is the derivative of the denominator (up to a constant), the integral is a logarithm:

如果分子是分母的导数（至多相差一个常数），则积分为对数函数：

∫ f′(x)⁄f(x) dx = ln|f(x)| + C

For more complicated denominators of the form (ax+b)(cx+d), use partial fraction decomposition:

对于形如 (ax+b)(cx+d) 的更复杂分母，可使用部分分式分解：

1⁄((ax+b)(cx+d)) = A⁄(ax+b) + B⁄(cx+d)

Find A and B by clearing denominators and comparing coefficients, then integrate each term separately.

通过去分母并比较系数求出 A 和 B，然后逐项积分。

10. Area Under a Curve | 曲线下的面积

The definite integral ∫ₐᵇ f(x) dx gives the signed area between the curve y = f(x) and the x-axis from x = a to x = b.

定积分 ∫ₐᵇ f(x) dx 表示曲线 y = f(x) 与 x 轴从 x = a 到 x = b 之间的有向面积。

If f(x) ≥ 0, the integral equals the actual area. If f(x) is negative, the integral is negative. To find total area, split the interval at points where f(x) = 0 and take absolute values.

若 f(x) ≥ 0，积分等于实际面积。若 f(x) 为负，积分为负。要求总面积，需要在 f(x) = 0 的点处分割区间并取绝对值。

For the area between two curves y = f(x) and y = g(x), where f(x) ≥ g(x), the formula is:

对于两条曲线 y = f(x) 与 y = g(x) 之间的面积，当 f(x) ≥ g(x) 时，公式为：

A = ∫ₐᵇ [f(x) − g(x)] dx

11. Volumes of Revolution | 旋转体的体积

When a region under a curve is rotated about an axis, the resulting solid has a volume that can be found by integration.

当曲线下的区域绕某轴旋转时，所得立体图形的体积可以通过积分求得。

For rotation about the x-axis, using discs:

绕 x 轴旋转时，使用圆盘法：

V = π∫ₐᵇ [f(x)]² dx

For rotation about the y-axis, using shells:

绕 y 轴旋转时，使用柱壳法：

V = 2π∫ₐᵇ x f(x) dx

These formulas require that the region being rotated is defined clearly. Always sketch the region and identify the axis of rotation.

这些公式要求明确旋转区域。务必先画出区域并确定旋转轴。

12. Approximate Integration | 近似积分

Sometimes an antiderivative is impossible to find analytically. In such cases, we use numerical methods to approximate definite integrals.

有时无法解析地求出原函数。在这种情况下，我们使用数值方法来近似定积分。

Trapezoidal rule: Tₙ = (b−a)⁄(2n) [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]

梯形法则：Tₙ = (b−a)⁄(2n) [f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]
Simpson’s rule: Sₙ = (b−a)⁄(3n) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)] for even n.

辛普森法则：Sₙ = (b−a)⁄(3n) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)]，其中 n 为偶数。

Simpson’s rule is generally more accurate than the trapezoidal rule for smooth functions.

对于光滑函数，辛普森法则通常比梯形法则更精确。

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Integration: Basic Concepts and Common Methods | 积分基本概念与常见方法