🔍 不定积分：谁的导数是它？🔍 Indefinite Integrals: Whose Derivative Is This?
上一片里，导数是“坡度”，带机器人找到了山顶。现在把机器倒过来转：如果只知道变化率，能不能反推出原来的函数？能——这门“倒带”的手艺，就叫不定积分。Last leaf, the derivative was the “slope” that guided the robot to its summit. Now spin the machine backwards: if you only know the rate of change, can you recover the original function? Yes — the art of rewinding is called the indefinite integral.
第 1 步：倒带的导数机器Step 1: Run the derivative machine backwards
求导是顺放：放进原函数，出来变化率。倒着走，行不行？Differentiation runs forwards: feed in a function, out comes its rate of change. Can we run it in reverse?
行——倒着走就是不定积分：已知变化率，找回原函数。例：谁的导数是 2x？用幂法则想一想，x² 的导数正好是 2x，所以答案是 x²。机器一按倒带键，2x 就变回了 x²。Yes — running backwards is the indefinite integral: given the rate of change, recover the function. Example: whose derivative is 2x? Power rule to the rescue — the derivative of x² is exactly 2x, so the answer is x². Press rewind, and 2x turns back into x².
等等——只有 x² 这一个答案吗？Wait — is x² the only answer?
第 2 步：别忘 +C——起点的任意高度Step 2: Don't forget +C — the any starting height
不止一个：x² + 3、x² − 5 的导数也都是 2x，因为常数求导会消失。所以答案写成 x² + C：C 是任意常数，代表“起点的任意高度”。一族曲线形状相同，只是起点高低不同——这正是“不定”两个字的来历。Not the only one: x² + 3 and x² − 5 also have derivative 2x, because constants vanish when you differentiate. So we write x² + C: C is an arbitrary constant, the “any starting height”. One family of curves with the same shape, only the starting heights differ — that is why it is called “indefinite”.
这招在真实世界里怎么用？How is this used in the real world?
第 3 步：从速度反推路程Step 3: From speed back to distance
测速仪记下每一刻的速度，倒带（积分）就能反推走过的路程：路程是速度的原函数。C 就是出发时的起点——补上起点，整条路线才唯一。导航和运动手环，都在悄悄做这件事。积分号 ∫ 的记号，不妨理解成一台“倒带机”的按钮。A speedometer records how fast you move at every instant; rewinding (integrating) recovers the distance travelled: distance is the antiderivative of speed. The C is where you set off — add the start point and the route becomes unique. Navigation apps and fitness bands do this quietly all the time. Think of the integral sign ∫ as the rewind button.
🎮 你来倒带（谁的导数是它？）🎮 Your turn: rewind the machine
道理讲完了。给导数选出原函数的主体，再决定常数怎么办——4 关全过，你就是倒带大师。Theory done. Pick the main part of the antiderivative, then decide what happens to the constant — clear all 4 levels and you are a rewind master.
一句话记住它：不定积分就是求导的倒带——给定变化率，找回原函数，只差一个任意常数 C。Remember it in one line: an indefinite integral rewinds differentiation — given the rate, recover the function, up to an arbitrary constant C.
不定积分是求导的逆运算：谁的导数是它，就找谁The indefinite integral inverts differentiation: find the function whose derivative this is答案别忘 +C：常数求导会消失，C 是起点的任意高度Always add + C: constants vanish under differentiation, and C is any starting height从速度反推路程、从变化率复原总量，都是这台倒带机Speed-to-distance and rate-to-total are both this rewind machine at work
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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