Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>If \(\int x^{26}(x-1)^{17}(5x-3)\, dx = \dfrac{x^{27}(x-1)^{18}}{k} + C\), where \(C\) is constant of integration, then the value of \(k\) is:</p>
<p>(a) 3</p>
<p>(b) 6</p>
<p>(c) 9</p>
<p>(d) 12</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand x^26(x-1)^17(5x-3) is the derivative of x^27(x-1)^18 divided by some constant. Use the product rule: d/dx[x^27(x-1)^18] = 27x^26(x-1)^18 + 18x^27(x-1)^17, then factor to match the given integrand.
<p><strong>Step 1:</strong> Assume the integral form suggests we differentiate x^27(x-1)^18 and compare.</p><p><strong>Step 2:</strong> Using product rule: d/dx[x^27(x-1)^18] = 27x^26(x-1)^18 + 18x^27(x-1)^17</p><p><strong>Step 3:</strong> Factor out x^26(x-1)^17: = x^26(x-1)^17[27(x-1) + 18x] = x^26(x-1)^17[27x - 27 + 18x] = x^26(x-1)^17(45x - 27)</p><p><strong>Step 4:</strong> Factor the coefficient: 45x - 27 = 9(5x - 3), so d/dx[x^27(x-1)^18] = 9x^26(x-1)^17(5x - 3)</p><p><strong>Step 5:</strong> Therefore: ∫x^26(x-1)^17(5x-3) dx = x^27(x-1)^18/9 + C</p><p>∴ <strong>k = 9</strong></p>
Correct Answer: C