Question

Let the domain of the function $f(x)=\log _{4}\left(\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)\right)$ be $(a, b)$ Then the value of the integral $\int\limits_{a}^{b} \frac{\sin ^{3} x}{\left(\sin ^{3} x+\sin ^{3}(a+ b- x)\right)} d x$ is equal to________.

Answer 1

For domain
$\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right) > 0$
$\log _{3}\left(18 x-x^{2}-77\right) > 1$
$18 x-x^{2}-77 > 3$
$x^{2}-18 x+80 < 0$
$x \in(8,10)$
$\Rightarrow a=8$ and $b=10$
$I=\int\limits_{a}^{b} \frac{\sin ^{3} x}{b \sin ^{3} x+\sin ^{3}(a+ b -x)} d x$
$I=\int\limits_{a}^{b} \frac{\sin ^{3}(a+ b- x)}{\sin ^{3} x+\sin ^{3}(a +b -x)}$
$2 I=(b-a) \Rightarrow I=\frac{b-a}{2}$
$(\because a=8 \text { and } b=10)$
$I=\frac{10-8}{2}=1$