Question Number 109169 by ZiYangLee last updated on 21/Aug/20 $$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{1}\centerdot\mathrm{1}!\:+\:\mathrm{2}\centerdot\mathrm{2}!\:+\:\mathrm{3}\centerdot\mathrm{3}!\:+…+{n}\centerdot{n}! \\ $$$$\mathrm{is}\: \\ $$ Answered by Dwaipayan Shikari last updated on 21/Aug/20 $$\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}{n}.{n}!=\underset{{n}=\mathrm{1}}…
Question Number 43629 by peter frank last updated on 12/Sep/18 $$\mathrm{If}\:\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{an}\:\mathrm{infinite}\:\mathrm{GP}\:\mathrm{be}\:\mathrm{3}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{of}\:\mathrm{its}\:\mathrm{term}\:\mathrm{is}\:\mathrm{also}\:\mathrm{3}, \\ $$$$\mathrm{then}\:\mathrm{its}\:\mathrm{first}\:\mathrm{term}\:\mathrm{and}\:\mathrm{common}\:\mathrm{ratio}\:\mathrm{are} \\ $$ Answered by $@ty@m last updated on 13/Sep/18 $${ATQ},…
Question Number 43630 by peter frank last updated on 12/Sep/18 $$\mathrm{If}\:\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$${x}^{\mathrm{3}} −\mathrm{12}{x}^{\mathrm{2}} +\mathrm{39}{x}−\mathrm{28}=\mathrm{0}\:\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP},\:\mathrm{then} \\ $$$$\mathrm{their}\:\mathrm{common}\:\mathrm{difference}\:\mathrm{will}\:\mathrm{be} \\ $$ Answered by $@ty@m last updated on…
Question Number 43627 by peter frank last updated on 12/Sep/18 $$\underset{\:\mathrm{0}} {\overset{\pi^{\mathrm{2}} /\mathrm{4}} {\int}}\:\frac{\mathrm{sin}\:\sqrt{{x}}}{\:\sqrt{{x}}}\:{dx}\:= \\ $$ Commented by math khazana by abdo last updated on…
Question Number 43628 by peter frank last updated on 12/Sep/18 $$\mathrm{Three}\:\mathrm{numbers}\:\mathrm{form}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{GP}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{middle}\:\mathrm{number}\:\mathrm{is}\:\mathrm{doubled},\:\mathrm{then}\:\mathrm{the} \\ $$$$\mathrm{new}\:\mathrm{numbers}\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP}.\:\mathrm{The}\:\mathrm{common}\:\mathrm{ratio} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{GP}\:\mathrm{is} \\ $$ Answered by $@ty@m last updated on…
Question Number 43626 by peter frank last updated on 12/Sep/18 $$\int\:\frac{{x}\:\mathrm{cos}\:{x}+\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{3}} {e}^{\mathrm{sin}\:{x}} +{x}^{\mathrm{2}} }}\:{dx}\:= \\ $$ Answered by MJS last updated on 15/Sep/18 $$\int\frac{\mathrm{1}+{x}\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{3}} \mathrm{e}^{\mathrm{sin}\:{x}}…
Question Number 43624 by peter frank last updated on 12/Sep/18 $$\underset{\:\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\frac{\mathrm{tan}^{−\mathrm{1}} {x}}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:= \\ $$ Commented by math khazana by abdo last updated…
Question Number 43625 by peter frank last updated on 12/Sep/18 $$\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}\:{dx}\:=\:\mathrm{2} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 13/Sep/18 $$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{4}}} \frac{\sqrt{{tanx}}\:}{{tanx}.{cos}^{\mathrm{2}}…
Question Number 43603 by peter frank last updated on 12/Sep/18 $$\mathrm{If}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{two}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{an}\:\mathrm{infinite} \\ $$$$\mathrm{GP}\:\mathrm{is}\:\mathrm{1}\:\mathrm{and}\:\mathrm{every}\:\mathrm{term}\:\mathrm{is}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{sum} \\ $$$$\mathrm{of}\:\mathrm{all}\:\mathrm{the}\:\mathrm{successive}\:\mathrm{terms},\:\mathrm{then}\:\mathrm{its} \\ $$$$\mathrm{first}\:\mathrm{term}\:\mathrm{is} \\ $$ Answered by $@ty@m last updated on…
Question Number 43602 by peter frank last updated on 12/Sep/18 $$\mathrm{If}\:\:{a},\:{b},\:{c}\:\:\mathrm{are}\:\mathrm{in}\:\mathrm{GP}\:\mathrm{and}\:\mathrm{log}_{{c}} {a},\:\mathrm{log}_{{b}} {c},\:\mathrm{log}_{{a}} {b} \\ $$$$\mathrm{are}\:\mathrm{in}\:\mathrm{AP},\:\mathrm{then}\:\mathrm{the}\:\mathrm{common}\:\mathrm{difference} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{AP}\:\mathrm{is} \\ $$ Answered by $@ty@m last updated…