Final answer:
To find the value of the definite integral ∫¹₂ (x³ - x²) dx, we can use the power rule of integration to evaluate each term separately and then subtract the results. Simplifying the expression, we find that the integral is equal to 13/4.
Step-by-step explanation:
To find the value of the definite integral ∫¹₂ (x³ - x²) dx, we can use the power rule of integration. The power rule states that the integral of x^n dx is equal to (x^(n+1))/(n+1), where n is any constant. Applying the power rule, we can rewrite the integral as (∫ x³ dx) - (∫ x² dx). Integrating each term separately, we get [(x^(3+1))/(3+1)] - [(x^(2+1))/(2+1)]. Evaluating this expression from 1 to 2, we get [(2^4)/(4)] - [(2^3)/(3)] - [(1^4)/(4)] + [(1^3)/(3)]. Simplifying this expression, we find that ∫¹₂ (x³ - x²) dx is equal to 13/4.