Indefinite Integrals | 不定积分

📚 Indefinite Integrals | 不定积分

Indefinite integration is a core topic in Edexcel A Level Mathematics. It reverses differentiation and produces a family of functions, not one single function.

不定积分是 Edexcel A Level 数学的核心内容。它是微分的逆运算，得到的是一族函数，而不是单一函数。

The notation ∫ f(x) dx asks: which functions have derivative f(x)? The answer always includes an arbitrary constant because the derivative of any constant is zero.

记号 ∫ f(x) dx 表示：哪些函数的导数等于 f(x)？答案总是包含任意常数，因为任何常数的导数都为零。

1. What is an Indefinite Integral? | 什么是不定积分

An indefinite integral is written as ∫ f(x) dx and represents all antiderivatives of f(x). It is called ‘indefinite’ because no limits of integration are given.

不定积分写作 ∫ f(x) dx，它表示 f(x) 的所有原函数。之所以称为“不定”，是因为没有给出积分上下限。

For example, since the derivative of x² is 2x, we can write ∫ 2x dx = x² + C. The added constant C is essential because many functions can have the same derivative.

例如，因为 x² 的导数是 2x，所以可以写成 ∫ 2x dx = x² + C。加上常数 C 非常重要，因为很多函数可以具有相同的导数。

2. The Constant of Integration | 积分常数

When differentiating, any constant term disappears. Therefore, when integrating, we must add an arbitrary constant C to account for every possible constant that was lost.

在求导时，任何常数项都会消失。因此，在积分时必须加上任意常数 C，以涵盖所有可能丢失的常数。

Graphically, y = x² + 1, y = x² − 3 and y = x² + 5 all have derivative dy/dx = 2x. An indefinite integral describes this entire family of parallel curves.

从图像上看，y = x² + 1、y = x² − 3 和 y = x² + 5 的导数都是 dy/dx = 2x。不定积分描述的就是这一整族平行曲线。

If a question gives extra information, such as a point on the original curve, you can find the specific value of C.

如果题目给出额外信息，例如原曲线上的一个点，就可以求出具体的 C 值。

3. Integrating Powers of x | 幂函数积分

The power rule for integration states: for n ≠ −1, you increase the power by 1 and divide by the new power.

幂函数积分法则为：当 n ≠ −1 时，先将指数加 1，再除以新的指数。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

Example: ∫ x³ dx = x⁴/4 + C and ∫ √x dx = ∫ x^(1/2) dx = (2/3)x^(3/2) + C.

例如：∫ x³ dx = x⁴/4 + C；∫ √x dx = ∫ x^(1/2) dx = (2/3)x^(3/2) + C。

If the power is a fraction or a negative value other than −1, the same rule applies, but you must be careful with arithmetic.

如果指数是分数或除 −1 以外的负数，同样适用该法则，但要特别注意计算。

4. The Reciprocal Function | 倒数函数积分

The power rule does not work when n = −1 because division by zero would occur. Instead, the reciprocal function has a special result.

当 n = −1 时，幂函数积分法则不可用，因为会出现除以零的情况。倒数函数有一个特殊结果。

∫ 1/x dx = ln|x| + C

The absolute value is included so the logarithm is defined for negative x as well. In many pure mathematics problems, you may see ln x if the domain is restricted to x > 0.

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