The expression ∫ 0 n x d x ∫ 0 n x d x , where x and x are integral and fractional part of x and n ∈ N , is equal to

Step-by-step solution

We have, ∫ 0 n x d x = ∫ 0 1 0 d x + ∫ 1 2 1 d x + ∫ 2 3 2 d x + … + ∫ n - 1 n n - 1 d x = 1 2 - 1 + 2 3 - 2 + 3 4 - 3 + … … + n - 1 n - n - 1 = 1 + 2 + 3 + … . + n - 1 = n n - 1 2 and ∫ 0 n x d x = ∫ 0 n x - x d x = n 2 ∴   ∫ 0 n x d x ∫ 0 n x d x = n - 1