(i) Express \(f(x)\) in the form \(A + \frac{B}{2x+1} + \frac{C}{x+2}\).

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

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

(ii) Integrate \(f(x)\):

\(\int f(x) \, dx = \int \left( 2 + \frac{1}{2x+1} - \frac{2}{x+2} \right) \, dx\)

= \(2x + \frac{1}{2} \ln |2x+1| - 2 \ln |x+2| + C\).

Evaluate from 0 to 4:

\(\left[ 2x + \frac{1}{2} \ln |2x+1| - 2 \ln |x+2| \right]_0^4\)

= \(\left( 8 + \frac{1}{2} \ln 9 - 2 \ln 6 \right) - \left( 0 + \frac{1}{2} \ln 1 - 2 \ln 2 \right)\)

= \(8 + \frac{1}{2} \ln 9 - 2 \ln 6 + 2 \ln 2\)

= \(8 + \frac{1}{2} \ln 9 - 2 \ln 3\)

= \(8 - \ln 3\).