Show-that-1-In-x-x-4-dx-1-9-

Question Number 163574 by Zaynal last updated on 08/Jan/22

S h o w t h a t;

\int_{1}^{\infty} \frac{I n x}{x^{4}} d x = \frac{1}{9}

Answered by Ar Brandon last updated on 08/Jan/22

\int_{1}^{\infty} \frac{\ln x}{x^{4}} d x = \frac{\partial}{\partial α} ∣_{α = 0} \int_{1}^{\infty} \frac{x^{α}}{x^{4}} d x = \frac{\partial}{\partial α} ∣_{α = 0} {[\frac{x^{α - 3}}{α - 3}]}_{1}^{\infty}

= \frac{\partial}{\partial α} ∣_{α = 0} \frac{1}{3 - α} = ∣_{α = 0} \frac{1}{{(3 - α)}^{2}} = \frac{1}{3^{2}} = \frac{1}{9}