Let \(f'(x) = {\sin ^3}(2x)\cos (2x)\). Find \(f(x)\), given that \(f\left( {\frac{\pi }{4}} \right) = 1\).

evidence of integration     (M1)

eg\(\,\,\,\,\,\)\(\int {f'(x){\text{d}}x} \)

correct integration (accept missing \(C\))     (A2)

eg\(\,\,\,\,\,\)\(\frac{1}{2} \times \frac{{{{\sin }^4}(2x)}}{4},{\text{ }}\frac{1}{8}{\sin ^4}(2x) + C\)

substituting initial condition into their integrated expression (must have \( + C\))     M1

eg\(\,\,\,\,\,\)\(1 = \frac{1}{8}{\sin ^4}\left( {\frac{\pi }{2}} \right) + C\)

Note: Award M0 if they substitute into the original or differentiated function.

recognizing \(\sin \left( {\frac{\pi }{2}} \right) = 1\)     (A1)

eg\(\,\,\,\,\,\)\(1 = \frac{1}{8}{(1)^4} + C\)

\(C = \frac{7}{8}\)    (A1)

\(f(x) = \frac{1}{8}{\sin ^4}(2x) + \frac{7}{8}\)     A1     N5

[7 marks]