The value of x > 1 satisfying the equation ∫ 1 x t log ⁡ t d t = 1 4 , is

Step-by-step solution

Consider that, I = ∫ 1 x t log ⁡ t d t = log ⁡ t ⋅ t 2 2 1 x - ∫ 1 x 1 t ⋅ t 2 2 d t = x 2 2 log ⁡ x - 1 2 t 2 2 1 x = x 2 2 log ⁡ x - 1 2 x 2 2 - 1 2 ⇒ 1 4 = x 2 2 log ⁡ x - 1 4 x 2 - 1 ⇒ 1 2 x 2 log ⁡ x - 1 4 x 2 = 0 (as x > 1 ) ⇒ x 2 2 log ⁡ x - 1 = 0 ⇒ 2 log ⁡ x - 1 = 0 ⇒ log ⁡ x = 1 2 ⇒ x = e 1 2 ⇒ x = e