Question Number 86668 by jagoll last updated on 30/Mar/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{P}\left(\mid\mathrm{x}\mid\:>\:\mathrm{1}\:\right)\:\mathrm{if}\:\mathrm{x}\:\mathrm{has}\:\mathrm{a}\:\mathrm{PDF}\:\mathrm{of} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\begin{cases}{\frac{\mathrm{1}}{\mathrm{4}}\:,\:\:−\mathrm{2}<\mathrm{x}<\mathrm{2}}\\{\mathrm{0}\:,\:\mathrm{elsewhere}}\end{cases} \\ $$ Commented by john santu last updated on 30/Mar/20 $$\Rightarrow\mathrm{P}\left(\mathrm{x}<−\mathrm{1}\:\cup\:\mathrm{x}>\mathrm{1}\right) \\ $$$$=\:\int_{−\mathrm{2}}…
Question Number 152201 by mnjuly1970 last updated on 26/Aug/21 $$ \\ $$$$\:\:\:…\mathrm{Integral}… \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).{tan}^{\:−\mathrm{1}} \left({cot}\left({x}\right)\right){dx}\overset{?} {=}\:\mathrm{0} \\ $$$$\:\:\:\:\:{proof}\:::\:…. \\ $$$$\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\pi} {ln}\:\left({sin}\left({x}\right)\:\right).\:{tan}^{\:−\mathrm{1}} \left(\:{tan}\left(\frac{\pi}{\mathrm{2}}\:−{x}\:\right)\right){dx}…
Question Number 21131 by Tinkutara last updated on 13/Sep/17 $$\mathrm{Figure}\:\mathrm{shows}\:\mathrm{a}\:\mathrm{small}\:\mathrm{bob}\:\mathrm{of}\:\mathrm{mass}\:{m} \\ $$$$\mathrm{suspended}\:\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\mathrm{on}\:\mathrm{a}\:\mathrm{thin}\:\mathrm{rod} \\ $$$$\mathrm{by}\:\mathrm{a}\:\mathrm{light}\:\mathrm{inextensible}\:\mathrm{string}\:\mathrm{of}\:\mathrm{length} \\ $$$${l}.\:\mathrm{The}\:\mathrm{rod}\:\mathrm{is}\:\mathrm{rigidly}\:\mathrm{fixed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{circular} \\ $$$$\mathrm{platform}.\:\mathrm{The}\:\mathrm{platform}\:\mathrm{is}\:\mathrm{set}\:\mathrm{into} \\ $$$$\mathrm{rotation}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{angular}\:\mathrm{speed} \\ $$$$\omega,\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{bob}\:\mathrm{loses}\:\mathrm{contact}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{vertical}\:\mathrm{rod},\:\mathrm{is} \\…
Question Number 152203 by peter frank last updated on 26/Aug/21 $$\mathrm{If}\:\mathrm{x}\:\mathrm{is}\:\mathrm{real}\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{2}+\mathrm{i}\right)^{\left(\mathrm{1}+\mathrm{3i}\right)\mathrm{x}} +\left(\mathrm{2}−\mathrm{i}\right)^{\left(\mathrm{1}−\mathrm{3i}\right)\mathrm{x}} \\ $$$$\mathrm{is}\:\mathrm{also}\:\mathrm{real} \\ $$ Commented by MJS_new last updated on 26/Aug/21…
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Question Number 21124 by Joel577 last updated on 13/Sep/17 Answered by Tinkutara last updated on 13/Sep/17 $${Here}\:{w}\:{can}\:{be}\:{taken}\:{as}\:−{x}+{iy}. \\ $$$$\therefore−{iw}={i}\left({x}−{iy}\right)={y}+{ix},\:{which}\:{is}\:{in}\:\mathrm{1}^{{st}} \\ $$$${quadrant}.\:{Hence}\:{w}\:{will}\:{be}\:{A}. \\ $$ Commented by…
Question Number 86657 by john santu last updated on 30/Mar/20 $$\mathrm{solve}\:\left(\mathrm{1}+\mathrm{x}^{\mathrm{3}} \right)\mathrm{dy}\:−\mathrm{x}^{\mathrm{2}} \:\mathrm{y}\:\mathrm{dx}=\mathrm{0} \\ $$$$\mathrm{y}\left(\mathrm{1}\right)\:=\:\mathrm{2} \\ $$ Answered by jagoll last updated on 30/Mar/20 $$\int\:\frac{\mathrm{dy}}{\mathrm{y}}\:=\:\int\:\frac{\mathrm{x}^{\mathrm{2}}…
Question Number 86655 by jagoll last updated on 30/Mar/20 $$\underset{{x}\rightarrow−\mathrm{1}^{+} } {\mathrm{lim}}\:\frac{\sqrt{\pi}\:−\sqrt{\mathrm{arc}\:\mathrm{cos}\:\mathrm{x}}}{\:\sqrt{\mathrm{x}+\mathrm{1}}} \\ $$ Answered by john santu last updated on 30/Mar/20 $$\mathrm{L}'\mathrm{hopital}\:\mathrm{rule} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}^{+}…
Question Number 21116 by mondodotto@gmail.com last updated on 13/Sep/17 Answered by Tinkutara last updated on 13/Sep/17 $$\mathrm{tan}\:\mathrm{A}=\frac{\mathrm{3}}{\mathrm{4}}\Rightarrow\mathrm{sin}\:\mathrm{A}=\frac{\mathrm{3}}{\mathrm{5}},\mathrm{cos}\:\mathrm{A}=\frac{\mathrm{4}}{\mathrm{5}} \\ $$$$\mathrm{cos}\:\mathrm{A}=\mathrm{2cos}^{\mathrm{2}} \:\frac{\mathrm{A}}{\mathrm{2}}−\mathrm{1} \\ $$$$\mathrm{cos}^{\mathrm{2}} \:\frac{\mathrm{A}}{\mathrm{2}}=\frac{\mathrm{1}+\mathrm{cos}\:\mathrm{A}}{\mathrm{2}}=\frac{\mathrm{9}}{\mathrm{10}}\Rightarrow\mathrm{cos}\:\frac{\mathrm{A}}{\mathrm{2}}=\frac{\mathrm{3}}{\:\sqrt{\mathrm{10}}} \\ $$$$\therefore\mathrm{sin}\:\frac{\mathrm{A}}{\mathrm{2}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{10}}}…
Question Number 152187 by Tawa11 last updated on 26/Aug/21 $$\mathrm{Please}\:\mathrm{formular}\:\mathrm{for}\:\:\:\:\:\Gamma\left(\frac{\mathrm{8}}{\mathrm{3}}\right) \\ $$ Commented by Tawa11 last updated on 26/Aug/21 $$\mathrm{Or}\:\mathrm{generally}\:\:\:\:\:\:\:\:\Gamma\left(\frac{\mathrm{x}}{\mathrm{y}}\right) \\ $$ Answered by Olaf_Thorendsen…