Question

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

Answer 1

$\because 18x–x^2–77>3$
$\Rightarrow x^2–18x+80<0$
$\Rightarrow x\in (8,10)$
$\therefore a=8$ and $b=10$
Now,
Let $I=\underset{a}{\overset{b}{\int }}\frac{{\sin }^{3}x}{({\sin }^{3}x+{\sin }^3(a+b-x\left)\right)}dx$
$I=\underset{a}{\overset{b}{\int }}\frac{{\sin }^3(a+b-x)}{\left({\sin }^3\right(a+b-x)+{\sin }^3(x\left)\right)}dx$
$\Rightarrow 2I=\underset{a}{\overset{b}{\int }}dx$
$\Rightarrow 2I=b-a=2$
$\therefore I=1$