Answer
$$ - \cos 2\theta - \sin 4\theta + 2$$
Work Step by Step
$$\eqalign{
& f\left( \theta \right) = 2\sin 2\theta - 4\cos 4\theta \cr
& {\text{find an antiderivative of }}f\left( \theta \right) \cr
& F\left( \theta \right) = 2\left( { - \frac{1}{2}\cos 2\theta } \right) - 4\left( {\frac{1}{4}\sin 4\theta } \right) + C \cr
& F\left( \theta \right) = - \cos 2\theta - \sin 4\theta + C \cr
& {\text{using the initial condition }}F\left( {\frac{\pi }{4}} \right) = 2 \cr
& 2 = - \cos 2\left( {\frac{\pi }{4}} \right) - \sin 4\left( {\frac{\pi }{4}} \right) + C \cr
& 2 = - \cos \left( {\frac{\pi }{2}} \right) - \sin \left( \pi \right) + C \cr
& 2 = - \left( 0 \right) - \left( 0 \right) + C \cr
& C = 2 \cr
& so, \cr
& = - \cos 2\theta - \sin 4\theta + 2 \cr} $$
