Evaluate: integral from 1 to 2 of (3x^2 - 2x) dx

Problem: Evaluate the definite integral $\int_{1}^{2} (3x^2 - 2x) \, dx$.

Steps

1. Apply the Power Rule for integration ($\int x^n \, dx = \frac{x^{n+1}}{n+1}$)
   $$\left[ 3 \cdot \frac{x^3}{3} - 2 \cdot \frac{x^2}{2} \right]_{1}^{2}$$
2. Simplify the expression
   $$\left[ x^3 - x^2 \right]_{1}^{2}$$
3. Apply the Fundamental Theorem of Calculus (evaluate at upper limit minus lower limit)
   $$(2^3 - 2^2) - (1^3 - 1^2)$$
4. Arithmetic
   $$(8 - 4) - (1 - 1)$$
5. Final subtraction
   $$4 - 0 = 4$$