Integral Calculus
Partial fractions of rational integrand; limit at infinity
Grade Class 12
Question:
Let $P(x)$ be a quadratic polynomial with $P(1)=-1$. If $\displaystyle\int\frac{P(x)\,dx}{(2x-3)^2(3x-2)^2}=\frac{1}{5}\ln|f(x)|+C$ where $f(x)$ is rational and $\lim_{x\to\infty}f(x)=\frac{4}{3}$, then $|f(1)|$ is
Step-by-Step Solution
Key Concept: Partial fraction: $P(x)/((2x-3)^2(3x-2)^2)=A/(2x-3)+B/(2x-3)^2+C/(3x-2)+D/(3x-2)^2$. Match $P(1)=-1$ and the log form; $B=D=0$ gives $P(x)$ linear combination. Then $f(1)$ from formula.
$|f(1)|=2$.
Correct Answer: 2