How do you figure out the antiderivative of (x+1)(2x-1)

Just FOIL and anti-differentiate the result using the Power Rule to yield:
∫ (x + 1)(2x - 1) dx
= ∫ (2x^2 - x + 2x - 1) dx
= ∫ (2x^2 + x - 1) dx
= 2 ∫ x^2 dx + ∫ x dx - ∫ dx
= (2/3)x^3 + (1/2)x^2 - x + C.

I hope this helps!

Expand and integrate:

∫(x + 1)(2x - 1) dx = ∫(2x² + x - 1) dx = 2x³/3 + x²/2 - x + C

By taking the integral, obviously.

Now, what is it that you don't understand and had to ask a question about?

int((x+1)*(2*x-1), x) = int((2*x^2 + x - 1), x) = (2/3)*x^3 + (1/2)*x^2 - x + C