If the integral ∫ ln ⁡ x x 3 d x = f x 4 x 2 + C , where f e = - 3 and C is the constant of integration, then the value of f e 2 is equal to

Step-by-step solution

∫ ln ⁡ x ⏟ I 1 x 3 ⏟ I I d x = ln ⁡ x - 1 2 x 2 - ∫ 1 x - 1 2 x 2 d x (Using integration by parts) = - ln ⁡ x 2 x 2 + 1 2 x - 2 - 2 + C = - 1 4 x 2 2 ln ⁡ x + 1 + C ⇒ f x = - 2 ln ⁡ x - 1 ∴ f e 2 = - 2 ln ⁡ e 2 - 1 = - 4 - 1 = - 5