(a) Express \(f(x)\) in the form \(\frac{A}{1+x} + \frac{Bx+C}{2+x^2}\).

Equating coefficients, we find \(A = 2\), \(B = -1\), and \(C = 0\).

Thus, \(f(x) = \frac{2}{1+x} - \frac{x}{2+x^2}\).

(b) Integrate \(f(x)\) to find \(\int_0^4 f(x) \, dx\).

\(\int \frac{2}{1+x} \, dx = 2 \ln(1+x)\).

\(\int \frac{-x}{2+x^2} \, dx = -\frac{1}{2} \ln(2+x^2)\).

Evaluate from 0 to 4:

\(2 \ln(5) - 2 \ln(1) - \frac{1}{2} \ln(18) + \frac{1}{2} \ln(2)\).

Simplifying gives \(\ln \left( \frac{25}{3} \right)\).