Which of the following statements is correct for the function g($\alpha$) for $\alpha$ $\in$ R such that

$$g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $$

$$g(\alpha ) = \int\limits_{\pi /6}^{\pi /3} {{{{{\sin }^\alpha }\left( {{\pi \over 2} - x} \right)} \over {{{\cos }^\alpha }\left( {{\pi \over 2} - x} \right)x + {{\sin }^\alpha }\left( {{\pi \over 2} - x} \right)}}dx} $$<br><br>$$ = \int\limits_{\pi /6}^{\pi /3} {{{{{\cos }^\alpha }x} \over {{{\sin }^\alpha }x + {{\cos }^\alpha }x}}dx} $$<br><br>$\therefore$ $$2.g(\alpha ) = \int\limits_{\pi /6}^{\pi /3} {{{si{n^\alpha }x + {{\cos }^\alpha }x} \over {{{\sin }^\alpha }x + {{\cos }^\alpha }x}}dx} = \int\limits_{\pi /6}^{\pi /3} {dx} = {\pi \over 3} - {\pi \over 6} = {\pi \over 6}$$<br><br>$\Rightarrow$ $g(\alpha ) = {\pi \over {12}}$ i.e. a constant function hence an even function.