Integrating with Fractions: Understanding the Solution for (x^2 - 2x)/2

[omf24]

Can someone please help me with this problem. Please see the attached image.

The integration when removing the min and max is also provided, how is the answer x²/2 - x³/6? Do we not do anything with the 1 - 1/2 part? and the +2 part? Are we only looking at (x²-2x)/2?

I appreciate the help.

 

[mathman]

The integrand is x - x²/2.

 

[HallsofIvy]

$$-\frac{1}{2}\left(x^2- 2x+ 2\right)= -\frac{1}{2}x^2+ x- 1$$
Adding 1 to that gives the integrand
$$-\frac{1}{2}x^2+ x$$
as mathman says.

 

[omf24]

Thanks, that is exactly what I needed to know.