Which of the following statements is correct for the function g(), for R, such that g() = …

Question

Which of the following statements is correct for the function $g(\alpha)$, for $\alpha\in\mathbb{R}$, such that $g(\alpha) = \displaystyle\int_{\pi/6}^{\pi/3}\frac{\sin^\alpha x}{\cos^\alpha x+\sin^\alpha x}\,dx$

Accepted Answer

$$g(\alpha ) = \int\limits_{\pi /6}^{\pi /3} {{{{{\sin }^\alpha }\left( {{\pi  \over 2} - x} \right)} \over {{{\cos }^\alpha }\left( {{\pi  \over 2} - x} \right)x + {{\sin }^\alpha }\left( {{\pi  \over 2} - x} \right)}}dx} $$

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

$$ \therefore $$ $$2.g(\alpha ) = \int\limits_{\pi /6}^{\pi /3} {{{si{n^\alpha }x + {{\cos }^\alpha }x} \over {{{\sin }^\alpha }x + {{\cos }^\alpha }x}}dx}  = \int\limits_{\pi /6}^{\pi /3} {dx}  = {\pi  \over 3} - {\pi  \over 6} = {\pi  \over 6}$$

$$ \Rightarrow $$ $$g(\alpha ) = {\pi  \over {12}}$$ i.e. a constant function hence an even function.

Solutions · 1