(i) To express \(f(x)\) in the given form, perform polynomial long division on \(\frac{x^3 - x - 2}{(x-1)(x^2+1)}\) to find \(A\), and then equate the numerator to find \(B, C,\) and \(D\).

Divide \(x^3 - x - 2\) by \((x-1)(x^2+1)\) to get a quotient of \(A = 1\) and a remainder.

Equate the remainder to \(B(x^2+1) + (Cx+D)(x-1)\) and solve for \(B, C,\) and \(D\):

\(B = -1, C = 2, D = 0\).

(ii) Integrate \(f(x) = 1 - \frac{1}{x-1} + \frac{2x}{x^2+1}\) from 2 to 3:

\(\int_2^3 \left( 1 - \frac{1}{x-1} + \frac{2x}{x^2+1} \right) \, dx = \left[ x - \ln|x-1| + \ln(x^2+1) \right]_2^3\).

Evaluate the definite integral:

\(\left[ 3 - \ln|3-1| + \ln(3^2+1) \right] - \left[ 2 - \ln|2-1| + \ln(2^2+1) \right] = 1\).