Antiderivatives of Absolute Value

[turophile]

The problem:

What are the antiderivatives of |x|?

My work so far:

Let x ? 0. Then the antiderivatives of |x| are (1/2)x[sup:12dtvqeq]2[/sup:12dtvqeq] + C.
Let x < 0. Then ...

My question:

If an antiderivative of a function is a function that gives the original function when the derivative of the antiderivative is taken, I'm having a hard time thinking of how to get x (if x ? 0) and - x (if x < 0) when taking a derivative. Or, does the sign of x even matter when the antiderivative squares x? If that's the case, is my solution for x ? 0 also the solution for x < 0?

 

[mmm4444bot]

We can get the sign by writing x^2 as x * |x|

 

[turophile]

So would the answer be [x * abs(x)]/2? I'm not sure, because the question wording implies there may be more than one antiderivative.

 

[galactus]

Yep, that's it. \(\displaystyle \frac{x|x|}{2}\).