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 $4$.