Question Number 218674 by Spillover last updated on 14/Apr/25 Answered by Nicholas666 last updated on 14/Apr/25 $$\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{x}^{\mathrm{4}} }{\:\sqrt{\mathrm{8}+\mathrm{2}{x}^{\mathrm{2}} −{x}^{\mathrm{4}} \:}}\right)\sqrt{\frac{\mathrm{2}+{x}^{\mathrm{2}} }{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}…
Question Number 218662 by Nicholas666 last updated on 14/Apr/25 $$\: \\ $$$$\:\:\:\:{Prove}:\:\:\:\:\underset{\mathrm{0}} {\int}^{\infty} \:\frac{{sin}\left({x}\right)}{{x}}\:{dx}\:=\:\frac{\pi}{\mathrm{2}} \\ $$$$ \\ $$ Answered by SdC355 last updated on 14/Apr/25…
Question Number 218713 by Nicholas666 last updated on 14/Apr/25 Commented by Nicholas666 last updated on 14/Apr/25 $$\:\:\:{Hello}\:{friends},\: \\ $$$$\:{let}'{s}\:{solve}\:{the}\:{integral}\:{problem} \\ $$ Commented by Nicholas666 last…
Question Number 218709 by Nicholas666 last updated on 14/Apr/25 $$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \:\frac{{d}\theta}{\:\sqrt{{sin}\theta\:+{cos}\theta}} \\ $$$$ \\ $$ Answered by Nicholas666 last updated on 14/Apr/25…
Question Number 218624 by Nicholas666 last updated on 13/Apr/25 $$ \\ $$$$\:\:\:\:{Prove};\:\underset{\frac{\mathrm{1}}{{e}}} {\int}^{{e}} \:\frac{{t}^{\mathrm{2}} }{{e}^{{t}^{\mathrm{2}\:} } }\:{dt}\:\leqslant\:\mathrm{1}−\:\frac{\mathrm{1}}{{e}^{\mathrm{2}\:\:} } \\ $$$$\:\:\:{e}−{the}\:{base}\:{of}\:{natural}\:{logarithm} \\ $$$$ \\ $$ Answered…
Question Number 218626 by Nicholas666 last updated on 13/Apr/25 $$ \\ $$$$\:{Prove}\:{that}\:{for}\:{all}\:{real}\:{numbers}\:{a}\:{and}\:{b} \\ $$$${with}\:{a}<{b},\:{the}\:{following}\:{inequality}\:{holds}; \\ $$$$\left(\int_{{a}} ^{{b}} \mathrm{1}\:{dx}\right)^{\mathrm{3}} \leqslant\:\left({b}−{a}\right)\left(\int_{{a}} ^{{b}} \left({x}−{a}+\mathrm{1}\right)^{\mathrm{2}} {dx}\right)\left(\int_{{a}\:} ^{{b}} \frac{\mathrm{1}}{\left({a}−{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx}\right)…
Question Number 218583 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\int_{\mathrm{0}\:} ^{\infty} \:{e}^{−{x}} \:\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{f}\left({n}\right)}{{n}}\:{sin}\left({nx}\right)\right){dx}\:=\mathrm{1} \\ $$$$ \\ $$ Answered by aleks041103 last…
Question Number 218582 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({ax}\right)\:−\:{cos}\left({bx}\right)}{{x}^{\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{\mathrm{2}}\:\mid{b}−{a}\mid\: \\ $$$$ \\ $$ Answered by aleks041103 last updated on…
Question Number 218599 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\infty} \boldsymbol{{x}}^{\boldsymbol{{s}}−\mathrm{1}} \:\underset{\boldsymbol{{n}}=\mathrm{1}\:} {\overset{\infty} {\prod}}\left(\mathrm{1}−\boldsymbol{{e}}^{−\boldsymbol{{nx}}} \right)^{−\mathrm{24}} \:\boldsymbol{{dx}} \\ $$$$ \\ $$ Commented by…
Question Number 218598 by Nicholas666 last updated on 12/Apr/25 $$ \\ $$$$\:\:\:\underset{\mathrm{0}} {\int}^{\infty} \:\frac{\boldsymbol{{x}}}{\boldsymbol{{sinh}}\left(\boldsymbol{{x}}\right)}\boldsymbol{{ln}}\left(\frac{\boldsymbol{{x}}}{\boldsymbol{{e}}^{\boldsymbol{{x}}} −\mathrm{1}}\right)\boldsymbol{{dx}}\: \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…