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Question-172984




Question Number 172984 by dragan91 last updated on 04/Jul/22
Answered by Mathspace last updated on 04/Jul/22
Υ_1 +Υ_2 =∫_0 ^1 ln((√x)+(√(x+1)))((√x)−(√(x+1)))dx  =∫_0 ^1 ln(x−x−1)dx  =∫_0 ^1 ln(−1)dx=∫_0 ^1 ln(e^(iπ) )dx  =iπ∫_0 ^1 dx=iπ
$$\Upsilon_{\mathrm{1}} +\Upsilon_{\mathrm{2}} =\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right)\left(\sqrt{{x}}−\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}−{x}−\mathrm{1}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(−\mathrm{1}\right){dx}=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({e}^{{i}\pi} \right){dx} \\ $$$$={i}\pi\int_{\mathrm{0}} ^{\mathrm{1}} {dx}={i}\pi \\ $$

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