(a) Express \(f(x)\) in the form \(\frac{Ax + B}{4 + x^2} + \frac{C}{1 + x}\).

Equating coefficients, we have:

\((Ax + B)(1 + x) + C(4 + x^2) = 5x^2 + x + 11\).

Solving, we find \(A = 2\), \(B = -1\), \(C = 3\).

Thus, \(f(x) = \frac{2x - 1}{4 + x^2} + \frac{3}{1 + x}\).

(b) Integrate each term separately:

\(\int \frac{2x - 1}{4 + x^2} \, dx = \frac{A}{2} \ln(4 + x^2) + \frac{B}{2} \arctan\left(\frac{x}{2}\right)\).

\(\int \frac{3}{1 + x} \, dx = C \ln(1 + x)\).

Substitute limits 0 and 2:

\(a \ln(4 + 4) + b \arctan\left(\frac{2}{2}\right) + c \ln(1 + 2) - [a \ln(4) + b \arctan(0) + c \ln(1)]\).

Calculate: \(\ln 54 - \frac{\pi}{8}\).