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