Indefinite Integration
Integration by differentiation/verification
Grade 12
Question:
<p>If \(\int x^{26}(x-1)^{17}(5x-3)\, dx = \frac{1}{k} x^{27}(x-1)^{18} + C\), then the value of \(k\) is:</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 up to a constant factor. Use the product rule: d/dx[x^27(x-1)^18] = 27x^26(x-1)^18 + 18x^27(x-1)^17, then factor out x^26(x-1)^17 to find the multiplying constant.
<p><strong>Step 1:</strong> Differentiate the right side to verify it matches the integrand.</p><p>d/dx[x^27(x-1)^18] = 27x^26(x-1)^18 + 18x^27(x-1)^17</p><p><strong>Step 2:</strong> Factor out x^26(x-1)^17 from both terms:</p><p>= x^26(x-1)^17[27(x-1) + 18x]</p><p>= x^26(x-1)^17[27x - 27 + 18x]</p><p>= x^26(x-1)^17[45x - 27]</p><p>= 9·x^26(x-1)^17[5x - 3]</p><p><strong>Step 3:</strong> Compare with the integrand x^26(x-1)^17(5x-3):</p><p>d/dx[x^27(x-1)^18] = 9·x^26(x-1)^17(5x-3)</p><p><strong>Step 4:</strong> Therefore:</p><p>∫x^26(x-1)^17(5x-3)dx = (1/9)x^27(x-1)^18 + C</p><p>∴ k = <strong>9</strong></p>
Correct Answer: 9