TITLE: Solve trigonometric integral $\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx $
QUESTION [14 upvotes]: Please help me to solve the following integral:
$$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx$$
I have tried a lot, but no results. I only transformed this integral to the following also not easy integral:
$$\int \frac{1}{(1+t^2)(1+t^{2014})} dt.$$

REPLY [5 votes]: Notice the following properties of limits, $$\color{blue}{\int_{-a}^{a}f(x)dx =2\int_{0}^{a}f(x)dx\quad \text{if} \quad f(-x)=f(x)}$$
$$\color{blue}{\int_{0}^{a}f(x)dx =\int_{0}^{a}f(a-x)dx}$$
Now, let $$I=\int_{-\pi/2}^{\pi/2}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx$$  $$\implies I=2\int_{0}^{\pi/2}\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx\tag 1$$
Using secong property, we get $$I=2\int_{0}^{\pi/2}\frac{\sin^{2014}\left(\frac{\pi}{2}-x\right)}{\sin^{2014}\left(\frac{\pi}{2}-x\right)+\cos^{2014}\left(\frac{\pi}{2}-x\right)}dx$$ 
$$\implies I=2\int_{0}^{\pi/2}\frac{\cos^{2014}x}{\sin^{2014}x+\cos^{2014}x}dx\tag 2$$ 
Now, adding eq(1) & (2), we get 
$$2I=2\int_{0}^{\pi/2}\left(\frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x}+\frac{\cos^{2014}x}{\sin^{2014}x+\cos^{2014}x}\right)dx$$  $$\implies I=\int_{0}^{\pi/2}\left(\frac{\sin^{2014}x+\cos^{2014}x}{\sin^{2014}x+\cos^{2014}x}\right)dx$$ $$=\int_{0}^{\pi/2}1. dx=[x]_{0}^{\pi/2}=\left[\frac{\pi}{2}-0\right]=\color{blue}{\frac{\pi}{2}}$$