How do I integrate 1-|x| from -1 to 1

integral[-1;1](1-|x|)dx=?

where |x| is the absolute value of x.

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I tried

|x|= x for x>=0
-x for x<0

so integral[-1;1](1-|x|)dx = integral[-1;0](1-x)dx + integral[0;1](1+x)dx =

= -1-1/2+1+1/2 = 0 (can not be 0, should be 1)

You are integrating a triangular region; you can find the area without using calculus:

integral[-1;1] (1 -|x|) dx = (1/2) 1 . 2 = 1.

However, if you wish to do it via calculus, then you're going the right way; split the integral into two parts, but remember to switch the limits on the negative part (you're going from right to left on the negative quadrant, left to right on the positive quadrant).
This will give you the right answer.

integral[-1;1](1-|x|)dx = integral[0;-1](1+x)dx + integral[0;1](1-x)dx
= 1/2 + 1/2
= 1.

integral[0;-1](1+x)dx + integral[0;1](1-x)dx
should be
integral[-1;0](1+x)dx + integral[0;1](1-x)dx

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