# 積分常數

## 簡介

$\int \cos(x)\,dx = \sin(x) + C.$

C即為積分常數，利用下式可以確認這些函數的確都是$\cos(x)$的反導數：

\begin{align} \frac{d}{dx}[\sin(x) + C] &= \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[C] \\ &= \cos(x) + 0 \\ &= \cos(x) \end{align}

## 積分常數的必要性

\begin{align} \int 2\sin(x)\cos(x)\,dx &=& \sin^2(x) + C &=& -\cos^2(x) + 1 + C \\ \int 2\sin(x)\cos(x)\,dx &=& -\cos^2(x) + C &=& \sin^2(x) - 1 + C \end{align}

## 不同反導數之間只差一個常數的原因

\begin{align} \int_a^x 0\,dt &= F(x)-F(a)\\ &= F(x)-C, \end{align}

$\int {1 \over x}\,dx = \begin{cases}\ln \left|x \right| + C^- & x < 0\\ \ln \left|x \right| + C^+ & x > 0 \end{cases}$

## 參考資料

1. ^ Stewart, James. Calculus: Early Transcendentals 6th. Brooks/Cole. 2008. ISBN 0-495-01166-5.
2. ^ Larson, Ron; Edwards, Bruce H. Calculus 9th. Brooks/Cole. 2009. ISBN 0-547-16702-4.
3. ^ Albert Tarantola, "Inverse Problems: Exercices. Chapter 8: The Derivative Operator, its Transpose, and its Inverse", 12 March 2007
4. ^ "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012