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Question Number 172984 by dragan91 last updated on 04/Jul/22

Answered by Mathspace last updated on 04/Jul/22

$${\mathrm{{\rm Y}}}_1+{\mathrm{{\rm Y}}}_2={\int }_0^{1}ln(\sqrt{x}+\sqrt{x+1})(\sqrt{x}-\sqrt{x+1})dx$$$$={\int }_0^{1}ln(x-x-1)dx$$$$={\int }_0^{1}ln(-1)dx={\int }_0^{1}ln\left(e^{i\pi }\right)dx$$$$=i\pi {\int }_0^{1}dx=i\pi$$

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