Question:
Which of the following statements is correct for the function $g(\alpha)$ for $\alpha \in R$ such that
$g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{a} x+\sin ^{\alpha} x} d x$
Correct Option: 4,
Solution:
$g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\sin ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{a} x\right)} \ldots . .(i)$
$g(\alpha)=\int_{\frac{\pi}{6}}^{\pi / 3} \frac{\cos ^{\alpha} x}{\left(\sin ^{\alpha} x+\cos ^{\circ} x\right)} \ldots(i i)$
$(1)+(2)$
$2 g(\alpha)=\frac{\pi}{6}$
$g(\alpha)=\frac{\pi}{12}$
Constant and even function
Due to typing mistake it must be bonus.
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