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I know I can integrate $|x|$ using the the sign function $\operatorname{sgn}(x)$ as $\int|x|dx=$$\frac{x^2}{2}\operatorname{sgn}(x)+C$ where $\operatorname{sgn}(x)=\frac{x}{|x|}=\frac{d}{dx}|x|$. But when I differentiate $\frac{x^2}{2}\operatorname{sgn}(x)=\frac{x^3}{2|x|}$ I get

$$\frac{d}{dx}\frac{x^3}{2|x|}=\frac{3x^2(2|x|)+x^3(2\operatorname{sgn}(x))}{(2|x|)^2}=\frac{6x^2|x|+\frac{2x^4}{|x|}}{4x^2}=\frac{8x^4}{4x^2|x|}=2x\operatorname{sgn}(x)=\frac{2x^2}{|x|}=(2|x|)\frac{|x|}{|x|}$$

Which implies $2|x|= |x|$, but we all know $1\ne2$. Where did I go wrong?

2 Answers 2

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Perhaps $\dfrac{3x^2(2|x|)+x^3(2\operatorname{sgn}(x))}{(2|x|)^2}$ should be $\dfrac{3x^2(2|x|)-x^3(2\operatorname{sgn}(x))}{(2|x|)^2}$?

The quotient rule gives the derivative of $f(x) = \dfrac{g(x)}{h(x)}$ as $f'(x) = \dfrac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}.$

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    You're right, I had the quotient rule wrong.2012-03-22
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We have that the sign function is $1$ for positive numbers and $-1$ for negative ones. So we can define its integral as $x$ for $x>0$ and $-x$ for $x<0$. This means that we can use the absolute value function:

$$\int \operatorname{sgn}(x)dx = |x|+C$$

Similarily

$$\int|x|dx = \frac{x|x|}{2}+C$$

If I'm not wrong, we can integrate the sign function $n+1$ times with $\dfrac{x^n|x|}{(n+1)!}+C$