The value of the integral ∫0∞ xlog⁡x1+x22dx is

∫0∞ xlog⁡x1+x22dx=∫01 xlog⁡x1+x22dx+∫1∞ xlog⁡x1+x22dxPut x=1/y in the second integral, so that dx=−1/y2dy. Ifx→∞ then y→0, and if x=1 then y=1.∴∫1∞ xlog⁡x1+x22dx=∫10 1⋅log⁡y−1y1+1y22−1y2dy=−∫01 ylog⁡y1+y22dy⇒∫0∞ xlog⁡x1+x22dx=∫01 xlog⁡x1+x22dx−∫01 xlog⁡x1+x22dx=0.