📚 Exercise 25A: Basic Integration Rules | 练习25A：基本积分法则
In IB Mathematics, integration is one of the core topics in calculus. Exercise 25A is designed to build your fluency in finding antiderivatives of elementary functions using basic integration rules. Whether you are studying Mathematics: Analysis and Approaches or Applications and Interpretation, a solid command of these rules is vital for solving definite integrals, area problems, and differential equations. This revision guide walks you through the key concepts and techniques that underpin Exercise 25A, with step-by-step examples and common pitfalls to avoid.
在IB数学中，积分是微积分的核心主题之一。练习25A旨在通过基本积分法则培养你求初等函数反导数的熟练度。无论你学习的是分析与方法还是应用与解读，扎实掌握这些法则对于解决定积分、面积问题和微分方程至关重要。本复习指南将带你梳理练习25A背后的关键概念和技巧，并提供逐步的例题以及需要避免的常见错误。
1. The Concept of Antiderivative | 反导数的概念
erivative of a function f(x) is another function F(x) whose derivative is f(x). That is, F' ntiderivatives. The collection of all antiderivatives is called the indefinite integral, denoted by ∫ f(x) dx = F(x) + C. Exercise 25A focuses on evaluating such indefinite integrals for basic functions.
函数 f(x) 的反导数是另一个函数 F(x)，使得 F'(x) = f(x)。求反导数的过程称为反微分法，最一般的反导数会包含一个任意常数 C，因为常数的导数为零。例如，若 f(x)=2x，则 F(x)=x² 是一个反导数，而 x²+5、x²-3 也是反导数。所有反导数的集合称为不定积分，记作 ∫ f(x) dx = F(x) + C。练习25A专注于对基本函数求这样的不定积分。
2. The Power Rule for Integration | 幂函数的积分法则
The most frequently used rule in Exercise 25A is the power rule for integration. For any real number n ≠ -1, the integral of xⁿ is given by:
练习25A中最常用的法则是幂函数的积分法则。对于任何实数 n ≠ -1，xⁿ 的积分由下式给出：
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
can factor it out first. For ∫ 4x² dx, write it as 4·∫ x² dx = 4·(x³/3) + C = (4/3)x³ + C.
这里，指数增加 1，并除以新的指数。例如，∫ x³ dx = x⁴/4 + C。务必要加上积分常数。如果变量有系数，你可以先把它提取出来。对于 ∫ 4x² dx，写为 4·∫ x² dx = 4·(x³/3) + C = (4/3)x³ + C。
xponent is a fraction or negative (other than -1), the rule still applies. For instance, ∫ √x 1/x² dx = ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C. Exercise 25A contains many such variations.
当指数为分数或负数（-1 除外）时，该法则同样适用。例如，∫ √x dx = ∫ x¹/² dx = x³/²/(3/2) + C = (2/3)x³/² + C，以及 ∫ 1/x² dx = ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C。练习25A中包含许多这样的变体。
3. Integrating Constant Functions | 常数函数的积分
= -3x + C. This rule is often tested in combination with other functions in Exercise 25A.
常数 k 的积分就是 kx
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