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

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 $\int_{a}^{b} \frac{\sin ^{3} x}{\left(\sin ^{3} x+\sin ^{3}(a+b-x)\right)} d x$ is equal to $.....$

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Answer

c
For domain

$\log _{5}\left(\log _{3}\left(18 x-x^{2}-77\right)\right)\,>\,0$

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

$18 x-x^{2}-77\,>\,3$

$x^{2}-18 x+80\,<\,0$

$x \in(8,10)$

$\Rightarrow a=8 \text { and } b=10$

$I=\int_{a}^{b} \frac{\sin ^{3} x}{\sin ^{3} x+\sin ^{3}(a+b-x)} d x$

$I=\int_{a}^{b} \frac{\sin ^{3}(a+b-x)}{\sin ^{3} x+\sin ^{3}(a+b-x)}$

$2 I=(b-a) \Rightarrow I=\frac{b-a}{2}(\because a=8$ and $b=10)$

$I=\frac{10-8}{2}=1$

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