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]