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

Question

Accepted Answer

"For domain

${\log _5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right) > 0$

${\log _3}(18x - {x^2} - 77) > 1$

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

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

$x \in (8,10)$

$\Rightarrow$ a = 8 and b = 10

$$I = \int\limits_a^b {{{{{\sin }^3}x} \over {{{\sin }^3}x + {{\sin }^3}(a + b - x)}}} dx$$

$$I = \int\limits_a^b {{{{{\sin }^3}x(a + b - x)} \over {{{\sin }^3}x + {{\sin }^3}(a + b - x)}}} $$

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

$I = {{10 - 8} \over 2} = 1$"

Let the domain of the function

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

Solution

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

Solution

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More Definite Integration questions

Drill 25 more like these. Every day. Free.

Let the domain of the function

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