find-0-x-2-logx-x-2-1-3-dx-

Question Number 148567 by mathmax by abdo last updated on 29/Jul/21

find \int_{0}^{\infty} \frac{x^{2} logx}{{(x^{2} + 1)}^{3}} dx

Answered by Ar Brandon last updated on 29/Jul/21

Ω = \int_{0}^{\infty} \frac{x^{2} \log x}{{(x^{2} + 1)}^{3}} d x

= \int_{0}^{1} \frac{x^{2} \log x}{{(x^{2} + 1)}^{3}} d x + \int_{1}^{\infty} \frac{x^{2} \log x}{{(x^{2} + 1)}^{3}} d x

= \int_{0}^{1} \frac{x^{2} \log x}{{(x^{2} + 1)}^{3}} d x - \int_{0}^{1} \frac{x^{2} \log x}{{(x^{2} + 1)}^{3}} d x = 0