We evaluate the integral using the Fundamental theorem of Calculus, by using and antiderivative of the funtion in the integrand;
An antiderivative of x^3 - 2 x is = 1/4 x^4 - x^2
So we evaluate this antiderivative at the two limits of integration:
At x = 1 the antiderivative becomes: 1/4 (1) - 1 = - 3/4
At x = - 1/2 the antiderivative becomes: 1/4 (-1/2)^4 - (-1/2)^2 = - 15/64
Now we subtract the evaluation at the upper limit minus the evaluation at the lower limit:
- 3/4 - (-15/64) = - 33/64
Allow me to show you the actual area we have calculated in a graph for the integration:
The curve in blue is the original function you provided : f(x) = x^3 - 2x
You can see that there is an area above the x axis that has been integrated and that gives as a result a positive number.
The area below the x axis is the part of the integral that provides the negative part which as you see is dominant in this calculation, therefore resulting in a negative final result.
Please, make sure you type -33/64 in the box provided.