Question

Let the domain of the function $f(x)={\log }_4\left({\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)\right)$ be $(a,b)$ Then the value of the integral $\underset{a}{\overset{b}{\int }}\frac{{\sin }^{3}x}{\left({\sin }^{3}x+{\sin }^3(a+b-x)\right)}dx$ is equal to________.

Answer 1

Answer: 1

For domain
${\log }_5\left({\log }_3\left(18x-x^2-77\right)\right)>0$
${\log }_3\left(18x-x^2-77\right)>1$
$18x-x^2-77>3$
$x^2-18x+80<0$
$x\in (8,10)$
$\Rightarrow a=8$ and $b=10$
$I=\underset{a}{\overset{b}{\int }}\frac{{\sin }^{3}x}{b{\sin }^{3}x+{\sin }^3(a+b-x)}dx$
$I=\underset{a}{\overset{b}{\int }}\frac{{\sin }^3(a+b-x)}{{\sin }^{3}x+{\sin }^3(a+b-x)}$
$2I=(b-a)\Rightarrow I=\frac{b-a}{2}$
$(\because a=8\text{ and }b=10)$
$I=\frac{10-8}{2}=1$