Hot integral

Let $n=2014$ . Then we have:

$\begin{aligned} I & = \int \cos ((n+2)x)\sin^n x \ dx \\ & = \int (\cos ((n+1)x)\cos x - \sin ((n+1)x) \sin x)\sin^nx \ dx \\ & = \int \left(\cos ((n+1)x)\sin^n \cos x - \sin ((n+1)x)\sin^{n+1}x\right) dx & \small \color{#3D99F6} \text{Note that }\frac d{dx} \sin^{n+1}x \cos((n+1)x) \\ & = \frac {\sin^{n+1}x \cos((n+1)x)}{n+1} + C & \small \color{#3D99F6} = (n+1)\sin^n x \cos x \cos((n+1)x) - (n+1)\sin^{n+1}x \sin((n+1)x) \end{aligned}$

Therefore,

$\begin{aligned} \int_\pi^{2\pi} f(x) \ dx & = \int_\pi^{2\pi} \frac {\sin^{n+1}x \cos((n+1)x)}{n+1} \ dx & & \small \color{#3D99F6} \text{Since }f \left(\frac {3\pi}2 - x\right) = - f \left(x-\frac {3\pi}2\right) \\ & = \boxed{0} \end{aligned}$