Let the domain of the function f x = log 4 log 5 log 3 18 x - x 2 - 77 be a , b . Then the value of the integral ∫ a b sin 3 x sin 3 x + sin 3 a + b - x is equal to _____.

Step-by-step solution

For domain log 5 log 3 18 x - x 2 - 77 > 0 log 3 18 x - x 2 - 77 > 1 ⇒ 18 x - x 2 - 77 > 3 ⇒ x 2 - 18 x + 80 < 0 ⇒ x ∈ 8 , 10 ⇒ a = 8 and b = 10 I = ∫ a b sin 3 x sin 3 x + sin 3 a + b - x d x Apply ∫ a b f x d x = ∫ a b f a + b - x d x I = ∫ a b sin 3 a + b - x sin 3 x + sin 3 a + b - x d x ⇒ 2 I = b - a ⇒ I = b - a 2 ∵   a = 8   &   b = 10 ⇒ I = 10 - 8 2 = 1