Question Number 220038 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{8}} \:+\:\mathrm{1}}}\:{dx} \\ $$$$\: \\ $$ Answered by MrGaster last updated on…
Question Number 219970 by Nicholas666 last updated on 04/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:{e}^{{x}^{\mathrm{2}} } \:{dx} \\ $$$$ \\ $$ Answered by MATHEMATICSAM last updated on 04/May/25…
Question Number 219911 by Nicholas666 last updated on 03/May/25 $$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \left(\underset{{n}\geqslant\mathrm{1}} {\sum}\:\frac{{sin}\left(\mathrm{2}\pi{nx}\right)}{{n}}\right)\frac{{dx}}{{x}^{{s}+\mathrm{1}} } \\ $$$$ \\ $$ Commented by MrGaster last updated…
Question Number 219864 by Nicholas666 last updated on 02/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\:{n}+\mathrm{1}} }{{x}+\mathrm{1}}\:{dx}\:<\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$ \\ $$ Answered by MrGaster last…
Question Number 219865 by Nicholas666 last updated on 02/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{ln}\Gamma\left({x}\right){dx}\:=\:{ln}\:\sqrt{\mathrm{2}\pi} \\ $$$$ \\ $$ Answered by MrGaster last updated…
Question Number 219866 by Nicholas666 last updated on 02/May/25 $$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi/\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}{x}−\mathrm{1}} \theta\:\mathrm{cos}\:^{\mathrm{2}{y}−\mathrm{1}} \theta\:{d}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}\right)+\Gamma\left({y}\right)}\:\:\:\:\: \\ $$$$\: \\ $$ Answered by MrGaster…
Question Number 219863 by Nicholas666 last updated on 02/May/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$$ \\ $$ Answered by MrGaster last…
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Question Number 219704 by Spillover last updated on 01/May/25 Answered by SdC355 last updated on 01/May/25 $$\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\:\mathrm{ln}\left(\frac{\mathrm{1}}{{z}}−\mathrm{1}\right)\:\mathrm{d}{z}=\mathrm{0}\: \\ $$$$\int_{\mathrm{0}} ^{\:\rho} \:+\int_{\:\rho} ^{\:\mathrm{1}} =\mathrm{0}\:,\:\left(\rho<\mathrm{1}\right)…
Question Number 219761 by Spillover last updated on 01/May/25 Answered by Spillover last updated on 01/May/25 Answered by mr W last updated on 01/May/25 $$\mathrm{1}+{x}+{x}^{\mathrm{2}}…