(i) Express \(f(x)\) in the form \(A + \frac{B}{x+1} + \frac{C}{2x-3}\).

Equating coefficients, we find:

\(A = 2\), \(B = -2\), \(C = -1\).

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

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

\(\int f(x) \, dx = \int \left(2 + \frac{-2}{x+1} + \frac{-1}{2x-3}\right) \, dx\).

This becomes:

\(2x - 2\ln|x+1| - \frac{1}{2}\ln|2x-3| + C\).

Evaluate from 2 to 6:

\(\left[2x - 2\ln|x+1| - \frac{1}{2}\ln|2x-3|\right]_2^6\).

Calculate:

At \(x = 6\): \(12 - 2\ln 7 - \frac{1}{2}\ln 9\).

At \(x = 2\): \(4 - 2\ln 3 - \frac{1}{2}\ln 1\).

Result: \(8 - \ln\left(\frac{49}{3}\right)\).