From the symmetry of the density function to the Y axis, we can know the mathematical expectation: E(X) = 0, without calculation.

If calculated:

E(X) = ∫(0，- 1)X( 1+X)dX+∫( 1，0) X( 1-X)dX

= 0.5X^2+X^3/3 |(0，- 1)+0.5x^2-x^3/3|( 1,0)

=- 1/2 + 1/3 + 1/2 - 1/3

= 0

That is E(X) = 0.