Definite Integration
Evaluation of Definite Integrals
Grade 12
Question:
<p>Let \(A = \begin{bmatrix} 0 & 1 & 1 \\ x & 0 & 1 \\ 2x & 4x & 0 \end{bmatrix}\) and \(f(x) = 4x^2 + 2x\). Which of the following is/are correct?</p>
<p>(a) \(\displaystyle\int_{-1}^{1} f(x)\,dx = \dfrac{8}{3}\)</p>
<p>(d) \(\displaystyle\int_{0}^{1} \dfrac{dx}{(2x+1)^2} = \dfrac{1}{3}\)</p>
<p>\(\displaystyle\int_{-1}^{1} f(x)\,dx = \dfrac{8}{3}\)</p>
<p>Graph of \(f(x)\) is of the form shown in option (b)</p>
<p>Graph of \(f(x)\) is of the form shown in option (c)</p>
<p>\(\displaystyle\int_{0}^{1} \dfrac{dx}{(2x+1)^2} = \dfrac{1}{3}\)</p>
Step-by-Step Solution
Key Concept: For option (a): recognize f(x) = 4x² + 2x is even in the x² term but odd in the 2x term, so integrate by splitting. For option (d): use substitution u = 2x+1 to convert the rational integral into a standard power form.
<p><strong>Option (a):</strong></p><p><strong>Step 1:</strong> Split f(x) = 4x² + 2x into even and odd parts.</p><p>∫₋₁¹ 4x² dx = 4[x³/3]₋₁¹ = 4(1/3 - (-1/3)) = 4(2/3) = 8/3</p><p>∫₋₁¹ 2x dx = 0 (odd function over symmetric interval)</p><p><strong>Step 2:</strong> Total = 8/3 + 0 = 8/3 ✓ <strong>CORRECT</strong></p><p><br><strong>Option (d):</strong></p><p><strong>Step 1:</strong> Use substitution u = 2x + 1, so du = 2dx, dx = du/2</p><p>When x = 0: u = 1; when x = 1: u = 3</p><p><strong>Step 2:</strong> ∫₀¹ dx/(2x+1)² = ∫₁³ (1/u²)·(du/2) = (1/2)∫₁³ u⁻² du</p><p><strong>Step 3:</strong> = (1/2)[-1/u]₁³ = (1/2)(-1/3 + 1) = (1/2)(2/3) = 1/3 ✓ <strong>CORRECT</strong></p><p>∴ Answer: A, D</p>
Correct Answer: A,D