# 積分常數

## 簡介

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

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

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

## 積分常數的必要性

{\displaystyle {\begin{aligned}\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{aligned}}}

## 不同反導數之間只差一個常數的原因

{\displaystyle {\begin{aligned}\int _{a}^{x}0\,dt&=F(x)-F(a)\\&=F(x)-C,\end{aligned}}}

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

## 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