Evaluate ∫−12 x3−xdx

f(x)=x3−x=xx2−1=x(x−1)(x+1)  Sign scheme of f(x) is as shown in the following figure.  From the sign scheme, we have ∫−12 x3−xdx=∫−10 x3−xdx+∫01 −x3−xdx+∫12 x3−xdx=x44−x22−10+x22−x4401+x44−x2212=−14−12+12−14+(4−2)−14−12=−14+12+12−14+2−14+12=114