This Year is 2017

Using the Beta function:

$\displaystyle\int_0^\frac{\pi}{2}\sin^\frac1{n}(x)\cos^\frac{-1}{n}(x)dx = \frac1{2}\beta(\frac{n+1}{2n},\frac{n-1}{2n}) = \frac{\Gamma(\frac{n+1}{2n})\Gamma(\frac{n-1}{2n})}{2\Gamma(\frac{n+1+n-1}{2n})}$

Using Euler's Reflection Formula: $\displaystyle\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}$

$\displaystyle\int_0^\frac{\pi}{2}\sin^\frac1{n}(x)\cos^\frac{-1}{n}(x)dx=\frac{\Gamma(\frac1{2n}+\frac1{2})\Gamma(\frac1{2}-\frac1{2n})}{2} = \frac{\pi}{2\sin(\pi( \frac1{2n}+\frac1{2}))}$

Sub in n = 2017 for $\displaystyle\boxed{\frac{\pi}{2\sin(\frac{1009\pi}{2017})}}$