\(\displaystyle
\eqalign{
& L = \mathop {\lim }\limits_{n \to \infty } \int_0^{{\pi \over 3}} {{{{{\sin }^n}(x)} \over {{{\sin }^n}(x) + {{\cos }^n}(x)}}} dx \cr
& L = \mathop {\lim }\limits_{n \to \infty } \int_0^{{\pi \over 3}} {{{{{\sin }^n}(x) + {{\cos }^n}(x) - {{\cos }^n}(x)} \over {{{\sin }^n}(x) + {{\cos }^n}(x)}}} dx = ? \cr} \)
I'm basically stuck quite at the beginning...I can only assume the integral can be solved without borders and then the limit applied.
Any help is welcome.
\eqalign{
& L = \mathop {\lim }\limits_{n \to \infty } \int_0^{{\pi \over 3}} {{{{{\sin }^n}(x)} \over {{{\sin }^n}(x) + {{\cos }^n}(x)}}} dx \cr
& L = \mathop {\lim }\limits_{n \to \infty } \int_0^{{\pi \over 3}} {{{{{\sin }^n}(x) + {{\cos }^n}(x) - {{\cos }^n}(x)} \over {{{\sin }^n}(x) + {{\cos }^n}(x)}}} dx = ? \cr} \)
I'm basically stuck quite at the beginning...I can only assume the integral can be solved without borders and then the limit applied.
Any help is welcome.