If $\int\left[\left(\frac{x^{-6}-64}{4+2x^{-1}+x^{-2}}\right)\left(\frac{x^2}{4-4x^{-1}+x^{-2}}\right) - \frac{4x^2(2x+1)}{(1-2x)}\right] dx$ is equal to $f(x)$ where $f(1) = 2$ then $f(3) = $
Step-by-Step Solution
Key Concept: Careful polynomial factorization and cancellation reduces a complex rational function to a simple linear integrand.
The integral $\int \left[\frac{1-(4x^2)}{x^6(4x^2+2x+1)(4x^2-4x+1)} + \frac{4x^2(2x+1)}{(1-2x)}\right] dx$ is simplified by factoring and partial fractions. The numerator $(1-2x)(1+2x)(4x^2-2x+1)(4x^2+2x+1)$ cancels appropriately, reducing the first integral to $\int \frac{(1+2x)(4x^2-2x+1)}{(2x-1)} dx$, which simplifies to $\int (2x+1) dx = x^2 + x + c$.
Correct Answer: 8