\(\int_{1}^{2} (2 + 2x - 3x^{2}) \mathrm {d} x\)

= \((2x + x^{2} - x^{3})|_{1}^{2}\)

= \((2(2) + 2^{2} - 2^{3}) - (2(1) + 1^{2} - 1^{3})\)

= \(0 - 2 = -2\)