Calculate $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\frac{\sin^{2014}x}{\sin^{2014}x + \cos^{2014}x}}dx$$ Integration by parts and variable substitutes don't seem to lead anywhere. Help would be appreciated.
EDIT: a more general version of this question is answered here: How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?
Hints: add and subtract $\cos^{2014}x$ in the numerator. Do a change of variables.
Full work: let $n = 2014$.
$$I=\int_{-\pi/2}^{\pi/2} \frac{\sin^n x + \cos ^n x}{\sin^n x + \cos^n x} dx - \int_{-\pi/2}^{\pi/2} \frac{\cos^n x}{\sin^n x + \cos^n x} dx = \pi - \int_{-\pi/2}^{\pi/2} \frac{\cos^n x}{\sin^n x + \cos^n x} dx $$
Now the rightmost integral is
$$\int_{0}^{\pi} \frac{\cos^n x}{\sin^n x + \cos^n x} dx$$
because the integrand is of period $\pi$. Doing a change of variables $u = x-\pi/2$, we get
$$I = \pi - I, \therefore I = \pi/2$$