Let the domain of the function f(x) = log₄(log₅(log₃(18x - x² - 77))) be (a, b). Then the value of the integral ∫ₐᵇ sin³x / (sin³x + sin³(a+b-x)) dx is equal to ________.
Let the domain of the function f(x) = log₄(log₅(log₃(18x - x² - 77))) be (a, b). Then the value of the integral ∫ₐᵇ sin³x / (sin³x + sin³(a+b-x)) dx is equal to ________.
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1 Answer
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log? (18x-x²-77)>0 ⇒ 18x-x²-77>1 ⇒ x²-18x+78<0. Roots are 9±√3.
log? (.)>0 ⇒ log? (.)>1 ⇒ 18x-x²-77>3 ⇒ x²-18x+80<0 ⇒ (x-8) (x-10)<0.
8Domain is (8,10). a=8, b=10.
I = ∫? ¹? sin³x/ (sin³x+sin³ (18-x)dx. Using King's property.
I = ∫? ¹? sin³ (18-x)/ (sin³ (18-x)+sin³x)dx.
2I = ∫? ¹? dx = 2. I=1.
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