Dear Luis,
I am not sure whether this is what you need.
sols = DSolve[{y'[x] == Piecewise[{{-x, x < 0}, {x, x >= 0}}], y[-1] == 7/2}, y[x], x]
Note, that the use of Piecewise, which works better than Abs here. You can then plot
Plot[y[x] /. sols, {x, -10, 10}]
Then you can integrate
Integrate[y[x] /. sols, {x, -2, 2}]
Cheers,
M.
