Basic Concepts of Indefinite Integrals

Definition of an indefinite integral

Basic definition

If $F'(x)=f(x)$, then $F(x)$ is an antiderivative of $f(x)$, and

$$\int f(x)\,dx = F(x) + C$$

where $C$ is an arbitrary constant (integration constant).

Antiderivative

If $F'(x)=f(x)$ on interval $I$, $F$ is an antiderivative of $f$ on $I$.

Properties:

1. If $F$ is one antiderivative, $F+C$ (any constant $C$) is also an antiderivative.
2. Any two antiderivatives of $f$ differ by a constant.

Geometric meaning

An indefinite integral represents a family of curves whose slope at $x$ equals $f(x)$ at $x$.
Example: $\int 2x\,dx = x^2 + C$ is the family $y=x^2+C$.

Relation between integration and differentiation

Basic relations

1. $\dfrac{d}{dx}\bigl[\int f(x)\,dx\bigr] = f(x)$
2. $\int f'(x)\,dx = f(x) + C$

Interpretations

• Integration is the inverse of differentiation.
• $C$ reflects the non-uniqueness of antiderivatives.
• Physically, integrals represent accumulation (e.g., displacement is the integral of velocity).

Meaning of the integration constant

• Needed because derivatives lose constant information.
• Geometrically shifts the family vertically.

Example: $x^2$, $x^2+1$, $x^2-3$ are all antiderivatives of $2x$ (parabolas shifted up/down).

练习题

练习 1

验证 $F(x) = x^3 + 2x$ 是 $f(x) = 3x^2 + 2$ 的原函数。

参考答案

思路：求 $F'(x)$ 并比对 $f(x)$。
答案：$F'(x)=3x^2+2=f(x)$，成立。

练习 2

求 $f(x) = 2x + 1$ 的所有原函数。

参考答案

答案：$\int (2x+1)dx = x^2 + x + C$。

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Summary

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\int f(x)\,dx$	数学符号	integral	不定积分表示一族原函数
$C$	符号	constant	积分常数

中英对照

中文术语	英文术语	音标	说明
原函数	antiderivative	/ˌæntaɪdɪˈrɪvətɪv/	导数等于被积函数的函数
不定积分	indefinite integral	/ɪnˈdɛfɪnət ˈɪntɪɡrəl/	原函数的集合，含常数 $C$
积分常数	integration constant	/ˌɪntɪˈɡreɪʃn ˈkɒnstənt/	反映原函数族的平移

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