If n 2 n + 1 ∫ 0 1 1 - x n 2 n d x = 1177 ∫ 0 1 1 - x n 2 n + 1 d x , then n ∈ N is equal to _______.

Step-by-step solution

Let I 1 = ∫ 0 1 1 - x n 2 n d x and I 2 = ∫ 0 1 1 - x n 2 n + 1 d x Now on solving I 2 we get, I 2 = ∫ 0 1 1 - x n 2 n + 1 · 1 d x Now on using by parts integration we get, ⇒ I 2 = 1 - x n 2 n + 1 . x 0 1 - ∫ 0 1 2 n + 1 1 - x n 2 n - n x n - 1 x d x ⇒ I 2 = - n 2 n + 1 I 2 - I 1 as  I 1 = ∫ 0 1 1 - x n 2 n d x ⇒ 2 n 2 + n + 1 I 2 = n 2 n + 1 I 1 Given  n 2 n + 1 ∫ 0 1 1 - x n 2 n d x = 1177 ∫ 0 1 1 - x n 2 n + 1 d x ⇒ I 1 I 2 = 2 n 2