Indefinite Integration
Integration by Substitution
Grade 12
Question:
<p><strong>Match the integrals in List-I with their correct evaluations in List-II:</strong></p>
<table border="1" cellpadding="4" style="border-collapse:collapse">
<tr><th>List-I (Integral)</th><th>List-II (Result)</th></tr>
<tr><td>(P) \(\displaystyle\int\frac{(x^2+x^3)\,dx}{5x^4+10x^5+10x^2}\)</td><td>(1) \(-\cot\theta-\frac{\cot^3\theta}{3}+C\)</td></tr>
<tr><td>(Q) \(\displaystyle\int\frac{(2x^3+3x^2)dx}{x^4+4x^2}\) (with \(3(2x^3+5x^4)\) numerator)</td><td>(2) \(\dfrac{-1}{x+1}+C\) [C is const.]</td></tr>
<tr><td>(R) \(\displaystyle\int\frac{(3x^2+5x^4)\,dx}{(x^3+x^5+1)^2}\) </td><td>(3) \(\dfrac{\sqrt{2x^2-2x+1}}{x}+C\)</td></tr>
<tr><td>(S) \(\displaystyle\int\frac{(2x^4+5x^3)\,dx}{(x^4+x^5+2)^2}\)</td><td>(4) \(\dfrac{-1}{x^3+x^5+1}+C\)</td></tr>
</table>
<li>P→4, Q→3, R→1, S→2</li>
<li>P→3, Q→4, R→2, S→1</li>
<li>A→Q, B→P,C; C→P,S; D→P,R</li>
<li>P→2, Q→1, R→4, S→3</li>
Step-by-Step Solution
Key Concept: For each rational integral, try the substitution u = denominator. If d(denominator)/dx \propto numerator, it becomes \intdu/u^2.
<p><strong>Key approach for each:</strong></p>
<p>(P): Numerator is proportional to d/dx(x³+x⁵+1)=3x²+5x⁴. So $\int\frac{d(x^3+x^5+1)}{(x^3+x^5+1)^2} = \frac{-1}{x^3+x^5+1}+C$ → matches (4).</p>
<p>(Q): Recognise the substitution t=1/x−x to reduce to a standard √(t²+const) form → matches (3).</p>
<p>(R): Matches result (1) after trig substitution.</p>
<p>(S): After u-substitution with the denominator, matches (2).</p>
<p>Answer: <strong>A→Q, B→P,C; C→P,S; D→P,R</strong></p>
Correct Answer: A→Q, B→P,C; C→P,S; D→P,R