Evaluate $\int^{1}_{0}$(3 - 2x)dx

Analyze the question: We need the definite integral of (3 − 2x) from x = 0 to x = 1.

Concept: For a definite integral, find the antiderivative, then compute F(upper) − F(lower).

Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Solution:

∫₀¹ (3 − 2x) dx = [3x − x²]₀¹

At x = 1: 3(1) − (1)² = 3 − 1 = 2

At x = 0: 3(0) − 0² = 0

Result: 2 − 0 = 2

Check against the options:

• A. "33m" is clearly a typing error and not a valid answer.
• B. 5: no.
• C. 2: matches our working.
• D. 6: no.

The correct answer is **C.

Tip: For a linear integrand like this, you can double-check using geometry. The function goes from 3 at x = 0 to 1 at x = 1, so the area is a trapezoid with average height (3 + 1)/2 = 2 and width 1, giving 2 × 1 = 2. Also, remember to subtract the lower-limit value, even when it is 0, because in other questions it often will not be.