Question Number 123495 by bemath last updated on 25/Nov/20 $$\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{x}−\mathrm{1}}{\left(\mathrm{2}−\sqrt{{x}}\right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)}\:{dx}\:? \\ $$ Answered by MJS_new last updated on 26/Nov/20 $$\int\frac{{x}−\mathrm{1}}{\left(\mathrm{2}+\sqrt{{x}}\right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)}{dx}=\int\frac{{dx}}{\left(\sqrt{{x}}−\mathrm{2}\right)\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)}=…
Question Number 57948 by maxmathsup by imad last updated on 14/Apr/19 $${let}\:{A}\left(\xi\right)\:=\int_{\xi} ^{\xi^{\mathrm{2}} } \:\:\:\:\frac{{arctan}\left(\mathrm{1}+\xi{t}\right)−\frac{\pi}{\mathrm{4}}}{\:\sqrt{\mathrm{2}+\xi{t}}−\sqrt{\mathrm{2}−\xi{t}}}\:{dt} \\ $$$${find}\:{lim}_{\xi\:\rightarrow\mathrm{0}} \:\:{A}\left(\xi\right)\:. \\ $$$$ \\ $$ Terms of Service…
Question Number 123462 by mnjuly1970 last updated on 25/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}… \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}}\:\overset{??} {=}\frac{\mathrm{11}\zeta\left(\:\mathrm{3}\:\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…………….. \\ $$ Answered by Lordose…
Question Number 123456 by mnjuly1970 last updated on 25/Nov/20 $$\:\:\:\:\:\:…\:{advanced}\:\:\:{calculus}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}:: \\ $$$$\:\:\:\:\:\:\:\:\:\phi=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\mathrm{H}_{{n}} }{{n}^{\mathrm{3}} }\:\right)\overset{???} {=}\frac{\pi^{\mathrm{4}} }{\mathrm{72}} \\ $$$$\:\:\:\:\:\:\:{note}:\:\mathrm{H}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+…+\frac{\mathrm{1}}{{n}} \\ $$…
Question Number 123452 by Eric002 last updated on 25/Nov/20 $$\int\frac{{dx}}{\:\left({x}^{\mathrm{2}} +{n}\right)\sqrt{{x}^{\mathrm{2}} +{a}}} \\ $$ Commented by MJS_new last updated on 25/Nov/20 $$\mathrm{depends}\:\mathrm{on}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:{a}\:\mathrm{and}\:{n}\:\mathrm{and}\:\mathrm{their} \\ $$$$\mathrm{relation}… \\…
Question Number 123454 by mnjuly1970 last updated on 25/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}… \\ $$$$\:\:\:\:\:{calculate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:\overset{???} {=}\int_{\mathrm{0}} ^{\:\infty} \sqrt{{x}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left({cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)\right){dx} \\ $$ Answered by Olaf…
Question Number 188982 by normans last updated on 10/Mar/23 Commented by normans last updated on 10/Mar/23 $${from}\:\:\boldsymbol{{FB}}\:{difficult}\:{problem} \\ $$ Commented by MJS_new last updated on…
Question Number 57900 by maxmathsup by imad last updated on 13/Apr/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{xt}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{t}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dt}…
Question Number 57899 by maxmathsup by imad last updated on 13/Apr/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{off}\:\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{3}} }\:\:{and}\:\int_{\mathrm{0}}…
Question Number 123386 by bemath last updated on 25/Nov/20 $$\:{Given}\: \\ $$$${f}\left({x}\right)=\left(\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} +\left(\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}\right){x}+\left(\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}\right)+\mathrm{1} \\ $$$${then}\:{the}\:{value}\:{of}\:{f}\left(\mathrm{4}\right)\:=\:… \\ $$ Answered by…