Evaluate $\int^{2}_{0}(x^3 + x^2)$dx.

Analyze the question: We need the definite integral of x³ + x² from x = 0 to x = 2.

Concept: Use the power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1). Then apply the limits: F(upper) − F(lower).

Solution:

∫₀² (x³ + x²) dx = [x⁴/4 + x³/3] from 0 to 2

At x = 2:

• 2⁴/4 = 16/4 = 4
• 2³/3 = 8/3

Sum = 4 + 8/3 = 12/3 + 8/3 = 20/3

At x = 0: the value is 0.

Result = 20/3 = 6 2/3

Tip: For definite integrals of polynomials, integrate each term, substitute the upper limit, and subtract the lower limit. When the lower limit is 0, you only need to evaluate the upper limit. Remember to convert improper fractions to mixed numbers when the options are given that way: 20/3 = 6 remainder 2, so 6⅔.