Let f x = x 1 + x n 1 n , x ∈ ℝ - - 1 , n ∈ ℕ , n > 2 . If f n x = ( f o f o f . . . . upto n times) x , then lim n → ∞ ∫ 0 1 x n - 2 f n x d x is equal to

Step-by-step solution

Given, f x = x 1 + x n 1 n ,   x ∈ ℝ - - 1 ,   n ∈ ℕ ,   n > 2 , f n x = ( f o f o f . . . . upto n times) x , Now finding f f x = f x 1 + f x n 1 n ⇒ f f x = x 1 + x n 1 n 1 + x 1 + x n 1 n n 1 n ⇒ f f x = x 1 + 2 x n 1 n Similarly, f n x = x 1 + n   x n 1 n Now solving the integral we get, I = ∫ 0 1 x n - 1 1 + n x n 1 n d x Now let, 1 + n x n = t n ⇒ n 2 x n - 1 d x = n t n - 1   d t I = ∫ 1 1 + n 1 n   1 n t n - 1   d t