Definite Integration
Integration
Grade Class 12
Question:
<div><p><strong>MATRIX MATCH TYPE QUESTION</strong></p><table><thead><tr><th>List-I</th><th>List-II</th></tr></thead><tbody><tr><td>(P) $\int \frac{x^2(x^6+x^5-1)dx}{(2x^6+3x^5+2)^2}$</td><td>(1) $-\frac{1}{3} \frac{1}{(x^3-x+1)} + C$</td></tr><tr><td>(Q) $\int \frac{(x^5+x^4+x^2)dx}{\sqrt{4x^7+5x^6+10x^4}}$</td><td>(2) $\frac{1}{2}(1+x^{-2}+x^{-5})^{-2} + C$</td></tr><tr><td>(R) $\int \frac{(2x^{12}+5x^9)dx}{(x^5+x^3+1)^3}$</td><td>(3) $-\frac{1}{6}(2x^3+3x^2+2x^{-3})^{-1} + C$</td></tr><tr><td>(S) $\int \frac{x^2-\frac{1}{3}}{(x^3-x+1)^2} dx$</td><td>(4) $x\left(\frac{x^3}{25} + \frac{x^2}{20} + \frac{1}{10}\right)^{\frac{1}{2}} + C$</td></tr></tbody></table></div>
(A) P → 4; Q → 2; R → 3; S → 1
(B) P → 3; Q → 4; R → 2; S → 1
(C) P → 1; Q → 3; R → 2; S → 4
(D) P → 3; Q → 1; R → 2; S → 4
Step-by-Step Solution
Key Concept: The problem involves evaluating indefinite integrals using substitution methods, often requiring algebraic manipulation of the integrand to identify a function and its derivative.
<div><p>For (P), rewrite the integrand as $\int \frac{x^2(x^6+x^5-1)}{(2x^6+3x^5+2)^2} dx$. By substituting $u = 2x^3+3x^2+2x^{-3}$, we can relate it to option (3). For (Q), factor out $x^7$ from the square root to simplify the integrand. For (R), divide numerator and denominator by $x^{15}$ to simplify. For (S), use the substitution $u = x^3-x+1$, then $du = (3x^2-1)dx$, which matches option (1). Matching these leads to (B).</p></div>
Correct Answer: 2