Question Number 172984 by dragan91 last updated on 04/Jul/22
Answered by Mathspace last updated on 04/Jul/22
$$\Upsilon_{\mathrm{1}} +\Upsilon_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right)\left(\sqrt{{x}}−\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}−{x}−\mathrm{1}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(−\mathrm{1}\right){dx}=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({e}^{{i}\pi} \right){dx} \\ $$$$={i}\pi\int_{\mathrm{0}} ^{\mathrm{1}} {dx}={i}\pi \\ $$