Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) , \(x > \frac{5}{2}\) . The graph of \(f\) passes through (\(4\), \(0\)) .

Find \(f(x)\) .

attempt to integrate which involves \(\ln \)     (M1)

eg   \(\ln (2x - 5)\) , \(12\ln 2x - 5\) , \(\ln 2x\)

correct expression (accept absence of \(C\))

eg   \(12\ln (2x - 5)\frac{1}{2} + C\) , \(6\ln (2x - 5)\)     A2

attempt to substitute (4,0) into their integrated f     (M1)

eg \(0 = 6\ln (2 \times 4 - 5)\) , \(0 = 6\ln (8 - 5) + C\)

\(C = - 6\ln 3\)     (A1)

\(f(x) = 6\ln (2x - 5) - 6\ln 3\) \(\left( { = 6\ln \left( {\frac{{2x - 5}}{3}} \right)} \right)\) (accept \(6\ln (2x - 5) - \ln {3^6}\) )     A1     N5

Note: Exception to the FT rule. Allow full FT on incorrect integration which must involve \(\ln\).

[6 marks]

While some candidates correctly integrated the function, many missed the division by \(2\) and answered \(12\ln \left( {2x - 5} \right)\) . Other common incorrect responses included \(\frac{{12x}}{{{x^2} - 5x}}\) and \( - 122{\left( {x - 5} \right)^{ - 2}}\) . Finding the constant of integration also proved elusive for many. Some either did not remember the \(+C\) or did not try to find its value, while others misunderstood the boundary condition and attempted to calculate the definite integral from \(0\) to \(4\).