COMEDK 2025 Afternoon Solved Paper

Concept:Use partial fraction decomposition to break the integrand into simpler fractions, then integrate term by term.Explanation:Start with the given integral: $\int \frac{x}{(x-1)(x-2)^2} dx$.Write the partial fraction form:$\frac{x}{(x-1)(x-2)^2} = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$.Multiply both sides by $(x-1)(x-2)^2$:$x = A(x-2)^2 + B(x-1)(x-2) + C(x-1)$.Expand and collect like terms:$x = (A+B)x^2 + (-4A-3B+C)x + (4A+2B-C)$.Equate coefficients of $x^2$, $x$, and constant term:$A+B = 0$,$-4A-3B+C = 1$,$4A+2B-C = 0$.Solve the system: from $A+B=0$, $B=-A$. Substituting gives $C = 1+A$ and $C = 2A$. Thus $2A = 1+A \Rightarrow A=1$, $B=-1$, $C=2$.So the integrand becomes:$\frac{1}{x-1} - \frac{1}{x-2} + \frac{2}{(x-2)^2}$.Integrate each term:$\int \frac{1}{x-1} dx = \ln|x-1|$,$\int -\frac{1}{x-2} dx = -\ln|x-2|$,$\int \frac{2}{(x-2)^2} dx = 2 \cdot \left(-\frac{1}{x-2}\right) = -\frac{2}{x-2}$.Combine the logarithms:$\ln|x-1| - \ln|x-2| = \ln\left|\frac{x-1}{x-2}\right|$.Thus the integral is:$\ln\left|\frac{x-1}{x-2}\right| - \frac{2}{x-2} + C$.Compare with the given form $a\ln\left|\frac{x-1}{x-2}\right| + \frac{b}{x-2} + c$.We get $a = 1$, $b = -2$.Answer:Option C: $a=1$, $b=-2$.