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Question Number 152533 by mnjuly1970 last updated on 29/Aug/21

Answered by Kamel last updated on 29/Aug/21

$$I\stackrel{t=\sqrt{x}}{=}2{\int }_0^{+\mathrm{\infty }}\frac{Ln(1+t)}{t(1+t)}dt=2{\int }_0^1\frac{Ln(1+t)}{t(1+t)}dt+2{\int }_0^1\frac{Ln(1+t)-Ln\left(t\right)}{1+t}dt$$$$=-2{Li}_2\left(1\right)-{Ln}^2\left(2\right)+{Ln}^2\left(2\right)-2{Li}_2(-1)$$$$=4\frac{{\pi }^2}{12}=\frac{{\pi }^2}{3}$$

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