Indefinite Integration
Integration of rational functions
Grade 12
Question:
<p><strong>Assertion (A):</strong> For <span class="math">\(-1 < a < 4\)</span>, the value of <span class="math">\((a + k)\)</span> is <span class="math">\(\frac{9}{2}\)</span>.</p><p><strong>Reason (R):</strong> The given integral reduces to the form <span class="math">\(\int \frac{f'(x)}{f(x)} dx\)</span> where <span class="math">\(f(x) = (x-1)^{5/2}\)</span>.</p>
<p>(A) Both A and R are true and R is the correct explanation of A</p>
<p>(B) Both A and R are true but R is NOT the correct explanation of A</p>
<p>(C) A is true but R is false</p>
<p>(D) A is false but R is true</p>
Step-by-Step Solution
Key Concept: Recognize when an integral can be reduced to the logarithmic form by identifying f'(x)/f(x) pattern
<p><strong>Step 1:</strong> The given integral can be reduced to <span class="math">$\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + c$</span></p><p><strong>Step 2:</strong> With <span class="math">$f(x) = (x-1)^{5/2}$</span>, this form applies for <span class="math">$-1 < a < 4$</span></p><p><strong>Step 3:</strong> From the integral evaluation, we get <span class="math">$a + k = \frac{9}{2}$</span></p><p>∴ Both A and R are true and R explains A correctly.</p>
Correct Answer: A