Let I n = ∫ 1 e x 19 ( log | x | ) n d x , where n ∈ N . If 20 I 10 = α I 9 + β I 8 , for natural numbers α and β , then α - β equal to _______.

Step-by-step solution

Given I n = ∫ 1 e x 19 ( log | x | ) n d x ,   n ∈ N ⇒ I n = ∫ 1 e x 19 log x n d x ,   ∵ for 1 < x < e ,   x > 0 Integrating by using by parts rule, ∫ a b u · v d x = u ∫ v d x a b - ∫ a b d d x u ∫ v d x d x , taking u = log x   &   v = x 19 , we get I n = log x n ∫ x 19 d x 1 e - ∫ 1 e d d x log x n ∫ x 19 d x d x ⇒ I n = log x n x 20 20 1 e - ∫ 1 e n log x n - 1 × 1 x × x 2