Let $f(x)$ is a quadratic function such that $f(0) = 1 \& f(-1) = 4$. If $\int \frac{f(x)dx}{x^2(x + 1)^2}$ is a rational function, then $f(10) = $
Step-by-Step Solution
Key Concept: For an integral to yield a rational function, the numerator polynomial must match the coefficients from partial fraction decomposition.
Given $g(x) = \int \frac{f(x)dx}{x^2(x+1)^2}$ is rational, use partial fractions: $\frac{f(x)}{x^2(x+1)^2} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x+1} + \frac{D}{(x+1)^2}$. For $g(x)$ to be rational, $A = C = 0$. Thus $f(x) = B(x+1)^2 + Dx^2$. Using $f(0) = 1$ gives $B = 1$; using $f(-1) = 4$ gives $D = 4$. Therefore $f(x) = 5x^2 + 2x + 1$ and $f(10) = 521$.
Correct Answer: 1,2,3