TITLE: Solve $\lim_{n\to \infty} \int_{0}^{\pi\over3} {{\sin^nx}\over \sin^nx+\cos^nx}dx$
QUESTION [8 upvotes]: $$\lim_{n\to \infty} \int_{0}^{\pi\over3} {{\sin^nx}\over \sin^nx+\cos^nx}dx$$
I really don't know what to do. It was really simple if the upper limit was $\pi/2$, because I can change $x=\pi/2-t$, but in my case I have no idea.
Some help please?  
Thank you!

REPLY [8 votes]: $$\lim_{n\to \infty} \int_{0}^{\pi\over3} {{\sin^nx}\over \sin^nx+\cos^nx}dx$$
$$=\lim_{n\to \infty} \int_{0}^{\pi\over3} \frac{1}{ \cot^n x+1} dx$$
We have that for $x \in (0,\frac{\pi}{4})$, that $1<\cot x$. Hence there $\frac{1}{ \cot^n x+1} \to 0$ as $n \to \infty$. 
We also have $0<\cot x<1$ for $x \in (\frac{\pi}{4},\frac{\pi}{3})$. Hence there $\frac{1}{ \cot^n x+1} \to 1$.
So we have,
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} 1 dx$$
$$=\frac{\pi}{12}$$