Definite Integration
Indefinite Integration
Grade Class 12
Question:
<p><strong>MATRIX MATCH TYPE QUESTION</strong></p><table style="width: 100%;"><tr><td><strong>List-I</strong></td><td><strong>List-II</strong></td></tr><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></table><p>(where <em>C</em> is the constant of integration.)</p>
(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: Use substitution method for indefinite integration by identifying the derivative of the denominator or a part of the integrand.
For (P), substitute $u = 2x^6+3x^5+2$. For (Q), factor out $x^7$ from the square root. For (R), divide numerator and denominator by $x^{15}$. For (S), substitute $u = x^3-x+1$. Matching these leads to (B).
Correct Answer: B