Let the domain of the function f (x) = _ 4 ( _ 5 ( _ 3 (18x - x ^ 2...

For domain $\log_{5}\left(\log_{3}\left(18x - {x}^{2} - 77\right)\right) > 0$

$\log_{3}\left(18x - {x}^{2} - 77\right) > 1$

$⇒ 18x - {x}^{2} - 77 > 3$

$⇒ {x}^{2} - 18x + 80 < 0$

$⇒ x \in \left(8, 10\right)$

$⇒ a = 8$ and $b = 10$

$I = {∫}_{a}^{b}\dfrac{\sin^{3}x}{\sin^{3}x + \sin^{3}\left(a + b - x\right)}dx$

Apply ${∫}_{a}^{b}f\left(x\right)dx = {∫}_{a}^{b}f\left(a + b - x\right)dx$

$I = {∫}_{a}^{b}\dfrac{\sin^{3}\left(a + b - x\right)}{\sin^{3}x + \sin^{3}\left(a + b - x\right)}dx$

$⇒ 2I = \left(b - a\right) ⇒ I = \dfrac{b - a}{2}\left(∵ a = 8 \text{ and } b = 10\right)$

$⇒ I = \dfrac{10 - 8}{2} = 1$