Let α and β α < β are roots of 18 x 2 - 9 π x + π 2 = 0 , f x = x 2 , g x = cos x . Then ∫ α β x g o f x d x =

Step-by-step solution

Given f x = x 2 ,   g x = cos x i.e. g f x = cos x 2 For 18 x 2 - 9 π x + π 2 = 0 roots are 9 π ± 81 π 2 - 72 π 2 36 = 9 π ± 3 π 36 = π 3 ,   π 6 i.e. α = π 6 ,   β = π 3 So, ∫ α β x g ∘ f x d x = ∫ π 6 π 3 x cos x 2 d x Let x 2 = t ;   x d x = d t 2 1 2 ∫ π 2 36 π 2 9 cos t d t = 1 2 sin π 2 9 - sin π 2 36