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Category: Differentiation

prove-that-0-sin-3-x-ln-x-x-dx-pi-8-2-ln-3-m-n-

Question Number 154478 by mnjuly1970 last updated on 18/Sep/21 $$ \\ $$$$ \\ $$$$\:\:{prove}\:{that}\:# \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{3}} \left(\:{x}\:\right).{ln}\left(\:{x}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\pi}{\mathrm{8}}\:\left(−\mathrm{2}\gamma\:+{ln}\left(\mathrm{3}\right)\right)\:…..\blacksquare\:{m}.{n}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\: \\ $$$$ \\…

Question-154058

Question Number 154058 by mnjuly1970 last updated on 13/Sep/21 Answered by qaz last updated on 14/Sep/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{xsin}\:\left(\mathrm{lnx}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$$$=−\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{e}^{−\mathrm{2u}} \mathrm{sin}\:\mathrm{u}}{\mathrm{1}+\mathrm{e}^{−\mathrm{2u}}…