The graph of the function \(y = f(x)\) passes through the point \(\left( {\frac{3}{2},4} \right)\) . The gradient function of f is given as \(f'(x) = \sin (2x - 3)\) . Find \(f(x)\) .

evidence of integration

e.g. \(f(x) = \int {\sin (2x - 3){\rm{d}}x} \)     (M1)

\( = - \frac{1}{2}\cos (2x - 3) + C\)     A1A1

substituting initial condition into their expression (even if C is missing)     M1

e.g. \(4 = - \frac{1}{2}\cos 0 + C\)

\(C = 4.5\)     (A1)

\(f(x) = - \frac{1}{2}\cos (2x - 3) + 4.5\)     A1     N5

[6 marks]

While most candidates realized they needed to integrate in this question, many did so unsuccessfully. Many did not account for the coefficient of x, and failed to multiply by \(\frac{1}{2}\). Some of the candidates who substituted the initial condition into their integral were not able to solve for "c", either because of arithmetic errors or because they did not know the correct value for \(\cos 0\) .