Indefinite Integration
Integration with functional equations
Grade None
Question:
<p>Let <em>f(x) = x³ + ax² + bx + c</em>. If <em>f(x) + f'(x) + f''(x) + f'''(x) = x³</em>, then find <em>g(x)</em> where <em>g(x) = ∫ f(x)/x³ dx</em> and <em>g(1) = 1</em>. Which of the following are correct?</p><p>(a) <em>a = -3</em> (b) <em>b = 0</em> (c) <em>g(x) = x - 3 ln x</em> (d) <em>f(x) = x³ - 3x²</em></p>
<p>\(a = -3\)</p>
<p>\(b = 0\)</p>
<p>\(g(x) = x - 3\ln x\)</p>
<p>\(f(x) = x^3 - 3x^2\)</p>
Step-by-Step Solution
Key Concept: Expand f(x) + f'(x) + f''(x) + f'''(x) by computing all derivatives, then match coefficients with x³ to find a, b, c. Use the initial condition g(1) = 1 to verify the integration result.
<p><strong>Step 1: Find derivatives of f(x)</strong></p><p>f(x) = x³ + ax² + bx + c</p><p>f'(x) = 3x² + 2ax + b</p><p>f''(x) = 6x + 2a</p><p>f'''(x) = 6</p><p><strong>Step 2: Compute f(x) + f'(x) + f''(x) + f'''(x)</strong></p><p>= x³ + ax² + bx + c + 3x² + 2ax + b + 6x + 2a + 6</p><p>= x³ + (a+3)x² + (b+2a+6)x + (c+b+2a+6)</p><p><strong>Step 3: Match with x³</strong></p><p>Comparing with x³:</p><p>• Coefficient of x²: a + 3 = 0 ⟹ <strong>a = -3</strong> ✓(a)</p><p>• Coefficient of x: b + 2a + 6 = 0 ⟹ b - 6 + 6 = 0 ⟹ <strong>b = 0</strong> ✓(b)</p><p>• Constant: c + b + 2a + 6 = 0 ⟹ c + 0 - 6 + 6 = 0 ⟹ c = 0</p><p><strong>Step 4: Find f(x)</strong></p><p>f(x) = x³ - 3x² ✓(d)</p><p><strong>Step 5: Integrate g(x)</strong></p><p>g(x) = ∫ f(x)/x³ dx = ∫ (x³ - 3x²)/x³ dx = ∫ (1 - 3/x) dx</p><p>= x - 3ln|x| + k</p><p><strong>Step 6: Use g(1) = 1</strong></p><p>g(1) = 1 - 3ln(1) + k = 1 + k = 1 ⟹ k = 0</p><p>∴ <strong>g(x) = x - 3ln x</strong> ✓(c)</p><p><strong>Answer: (a), (b), (c), (d) are all correct</strong></p>
Correct Answer: ABCD