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 $$\int\frac{\mathrm{1}+{x}\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{3}} \mathrm{e}^{\mathrm{sin}\:{x}}…
Question Number 43625 by peter frank last updated on 12/Sep/18 $$\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 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 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…
Question Number 43374 by peter frank last updated on 10/Sep/18 $$\mathrm{A}\:\mathrm{drawer}\:\mathrm{contains}\:\mathrm{5}\:\mathrm{brown}\:\mathrm{socks}\:\mathrm{and}\:\mathrm{4} \\ $$$$\mathrm{blue}\:\mathrm{socks}\:\mathrm{well}\:\mathrm{mixed}.\:\mathrm{A}\:\mathrm{man}\:\mathrm{reaches} \\ $$$$\mathrm{the}\:\mathrm{drawer}\:\mathrm{and}\:\mathrm{pulls}\:\mathrm{out}\:\mathrm{2}\:\mathrm{socks}\:\mathrm{at} \\ $$$$\mathrm{random}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that} \\ $$$$\mathrm{they}\:\mathrm{match}? \\ $$ Answered by tanmay.chaudhury50@gmail.com last…
Question Number 43344 by peter frank last updated on 10/Sep/18 $$\mathrm{The}\:\mathrm{number}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:…,\:{n}\:\:\mathrm{are}\:\mathrm{arranged} \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{random}\:\mathrm{order}.\:\mathrm{The}\:\mathrm{probability} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:…,\:{k}\:\left({k}>{n}\right)\:\mathrm{appears} \\ $$$$\mathrm{as}\:\mathrm{neighbours}\:\mathrm{is} \\ $$ Commented by MrW3 last updated on…
Question Number 108790 by saorey0202 last updated on 19/Aug/20 $$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:\theta\:\mathrm{lying}\:\mathrm{between}\:\mathrm{0}\:\mathrm{and} \\ $$$$\frac{\pi}{\mathrm{2}}\:\mathrm{and}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\begin{vmatrix}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \theta}&{\:\:\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{4}\:\mathrm{sin}\:\mathrm{4}\theta}\\{\:\:\:\mathrm{sin}^{\mathrm{2}} \theta}&{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{4}\:\mathrm{sin}\:\mathrm{4}\theta}\\{\:\:\:\mathrm{sin}^{\mathrm{2}} \theta}&{\:\:\:\mathrm{cos}^{\mathrm{2}} \theta}&{\mathrm{1}+\mathrm{sin}^{\mathrm{4}} \theta}\end{vmatrix}=\mathrm{0}\:\:\mathrm{are} \\ $$ Commented by…
Question Number 43180 by gunawan last updated on 08/Sep/18 $$\mathrm{The}\:\mathrm{smallest}\:\mathrm{and}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{values}\:\mathrm{of} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)\:,\:\:\mathrm{0}\leqslant\:{x}\:\leqslant\:\mathrm{1}\:\:\mathrm{are} \\ $$ Commented by maxmathsup by imad last updated on 08/Sep/18 $${let}\:\varphi\left({x}\right)={arctan}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)\:\:\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{changement}\:{x}\:={cos}\theta\:{give}…
Question Number 43179 by gunawan last updated on 08/Sep/18 $$\mathrm{The}\:\mathrm{solution}\:\mathrm{of} \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}{a}}{\mathrm{1}+{a}^{\mathrm{2}} }\right)−\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{1}−{b}^{\mathrm{2}} }{\mathrm{1}+{b}^{\mathrm{2}} }\right)=\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}−{x}^{\mathrm{2}} }\right)\:\mathrm{is} \\ $$ Answered by tanmay.chaudhury50@gmail.com last…