Definite Integration
Properties of definite integrals
Grade 12
Question:
<p>The value of \(\int_0^2 (px^3 + qx + r)\, dx\), where \(p, q, r\) are constants, depends on the value of</p>
<p>(a) \(q\)</p>
<p>(b) \(r\)</p>
<p>(c) \(p\)</p>
<p>(d) \(p\) and \(t\)</p>
Step-by-Step Solution
Key Concept: When integrating a polynomial, the definite integral ∫₀² (px³ + qx + r)dx = [px⁴/4 + qx²/2 + rx]₀² evaluates to 4p + 2q + 2r. The integral depends only on the coefficients p, q, r of the polynomial terms, not on their individual values separately.
<p><strong>Step 1:</strong> Apply the power rule and integrate each term:</p><p>∫₀² (px³ + qx + r)dx = [p·x⁴/4 + q·x²/2 + r·x]₀²</p><p><strong>Step 2:</strong> Evaluate at the limits:</p><p>= [p(2)⁴/4 + q(2)²/2 + r(2)] - [0]</p><p>= [16p/4 + 4q/2 + 2r]</p><p>= 4p + 2q + 2r</p><p><strong>Step 3:</strong> Interpret the result:</p><p>The definite integral value depends on the specific values of all three constants <strong>p, q, and r</strong> combined in the expression 4p + 2q + 2r.</p><p>∴ Answer: B (all three constants p, q, and r)</p>
Correct Answer: B