If $\int \frac{(x^2 + 2)dx}{(x^2 + 1)(x^2 + 4)} = k\tan^{-1}\left(\frac{mx}{c - x^2}\right)$, then find the value of $3k + m + c$.
Step-by-Step Solution
Key Concept: For a proper rational function with distinct quadratic factors in the denominator, use partial fractions to split into simpler terms with linear numerators.
Decompose the partial fraction $\frac{x^2 + 2}{(x^2+1)(x^2+4)} = \frac{1}{3(x^2+1)} + \frac{1}{3(x^2+4)}$. This is verified by the cover-up method: $P = 2$, $Q = 4$, and $R = 4$, giving the coefficients in the decomposition.
Correct Answer: 1,2,3