Question Number 223728 by Nicholas666 last updated on 03/Aug/25
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}_{\:} } ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\sqrt{{x}}\right)\centerdot\mathrm{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+\sqrt{{x}}}\:\mathrm{d}{x} \\ $$
Answered by Tawa11 last updated on 14/Aug/25
$$\mathrm{Finally}: \\ $$$$\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}\:+\:\sqrt{\mathrm{x}}\right)\:\mathrm{ln}\left(\mathrm{1}\:\:+\:\:\mathrm{x}\right)}{\mathrm{1}\:\:+\:\:\sqrt{\mathrm{x}}}\:\mathrm{dx} \\ $$$$\:\:=\:\:\frac{\mathrm{7}}{\mathrm{2}}\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{2}\right)\:\:−\:\:\mathrm{10}\:\mathrm{ln}\left(\mathrm{2}\right)\:\:−\:\:\frac{\pi^{\mathrm{2}} }{\mathrm{24}}\:\:\:+\:\:\mathrm{8}\:\:−\:\:\pi\:\:+\:\:\frac{\pi}{\mathrm{2}}\:\mathrm{ln}\left(\mathrm{2}\right)\:\:−\:\:\frac{\mathrm{11}}{\mathrm{12}}\:\mathrm{ln}^{\mathrm{3}} \left(\mathrm{2}\right)\:\:+\:\:\frac{\pi^{\mathrm{2}} }{\mathrm{48}}\:\mathrm{ln}\left(\mathrm{2}\right) \\ $$
Commented by Nicholas666 last updated on 17/Aug/25
$$\mathrm{wow}\:,\:\mathrm{but}\:\mathrm{can}\:\mathrm{you}\:\mathrm{show}\:\mathrm{that}? \\ $$
Commented by Tawa11 last updated on 20/Aug/25
$$\mathrm{Yes}\:\mathrm{sir}. \\ $$$$\mathrm{But}\:\mathrm{long}\:\mathrm{derivation}. \\ $$