(a) Express \(f(x)\) in the form:

\(\frac{A}{1+2x} + \frac{B}{2-x} + \frac{C}{(2-x)^2}\)

Using partial fraction decomposition, equate:

\(2x^2 + 17x - 17 = A(2-x)^2 + B(1+2x)(2-x) + C(1+2x)\)

Expand and compare coefficients to find:

\(A = -4\), \(B = -3\), \(C = 5\).

(b) Integrate each term separately:

\(\int \frac{-4}{1+2x} \, dx = -2 \ln(1+2x)\)

\(\int \frac{-3}{2-x} \, dx = 3 \ln(2-x)\)

\(\int \frac{5}{(2-x)^2} \, dx = \frac{5}{2-x}\)

Substitute limits from 0 to 1:

\(-2 \ln(3) + 3 \ln(1) + \frac{5}{1} - (-2 \ln(1) + 3 \ln(2) + 5/2)\)

Simplify to obtain:

\(\frac{5}{2} - \ln 72\).