Question Number 150840 by mathdanisur last updated on 15/Aug/21 $$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{arctan}\left(\mathrm{x}\right)}{\mathrm{x}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)}\:\mathrm{dx}\:=\:? \\ $$ Answered by mathmax by abdo last updated on 17/Aug/21 $$\mathrm{f}\left(\mathrm{a}\right)=\int_{\mathrm{0}}…
Question Number 150841 by mathdanisur last updated on 15/Aug/21 $$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{a}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{b}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 15/Aug/21…
Question Number 150838 by mathdanisur last updated on 15/Aug/21 Answered by amin96 last updated on 15/Aug/21 $$\frac{\left(\mathrm{3}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{5}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{7}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)}{\left(\mathrm{4}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{6}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)\left(\mathrm{8}^{\mathrm{4}} +\frac{\mathrm{1}}{\mathrm{4}}\right)}=\frac{\left(\left(\mathrm{3}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} −\mathrm{9}\right)\left(\left(\mathrm{5}^{\mathrm{2}}…
Question Number 150839 by mathdanisur last updated on 15/Aug/21 $$\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{for}\:\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$ Answered by peter frank last updated on 15/Aug/21 $${e}^{{x}}…
Question Number 19741 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{and}\:\mathrm{arg}\left(\frac{{z}\:−\:\mathrm{2}}{{z}\:+\:\mathrm{2}}\right)\:=\:\frac{\pi}{\mathrm{6}},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}. \\ $$ Commented by Tinkutara last updated on 15/Aug/17 $$\mathrm{I}\:\mathrm{found}\:\mathrm{the}\:\mathrm{correct}\:\mathrm{equation}\:\mathrm{but}\:\mathrm{in}\:\mathrm{book} \\ $$$$“\mathrm{major}\:\mathrm{arc}''\:\mathrm{is}\:\mathrm{written}\:\mathrm{along}\:\mathrm{with}\:\mathrm{that} \\…
Question Number 19739 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{z}\:=\:{x}\:+\:{iy}\:\mathrm{is}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{number} \\ $$$$\mathrm{satisfying}\:\mid{z}\:+\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} \:=\:\mid{z}\:−\:\frac{{i}}{\mathrm{2}}\mid^{\mathrm{2}} ,\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 15/Aug/17…
Question Number 19740 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:\mid{z}^{\mathrm{2}} \:−\:\mathrm{1}\mid\:=\:\mid{z}\mid^{\mathrm{2}} \:+\:\mathrm{1},\:\mathrm{then}\:{z}\:\mathrm{lies}\:\mathrm{on} \\ $$ Answered by ajfour last updated on 16/Aug/17 $$\:\mid\left(\mathrm{x}+\mathrm{iy}\right)^{\mathrm{2}} −\mathrm{1}\mid=\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{1}…
Question Number 19738 by Tinkutara last updated on 15/Aug/17 $$\mathrm{Locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{point}\:{z}\:\mathrm{satisfying}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid{iz}\:−\:\mathrm{1}\mid\:+\:\mid{z}\:−\:{i}\mid\:=\:\mathrm{2}\:\mathrm{is} \\ $$ Commented by math khazana by abdo last updated on 22/Jun/18 $${let}\:{z}\:={x}+{iy}\:\:\left({e}\right)\:\Leftrightarrow\mid{ix}−{y}−\mathrm{1}\mid\:+\mid{x}+{i}\left({y}−\mathrm{1}\right)\mid=\mathrm{2}…
Question Number 150804 by mathdanisur last updated on 15/Aug/21 Answered by aleks041103 last updated on 15/Aug/21 $${Let}\:{P}=\left({b}−{a}\right)\left({c}−{a}\right)\left({d}−{a}\right)\left({c}−{b}\right)\left({d}−{b}\right)\left({d}−{c}\right). \\ $$$${Obviously}\:{P}\:\:{is}\:{the}\:{product}\:{of}\:{all} \\ $$$${possible}\:{differences}\:{between}\:{a},{b},{c},{d}. \\ $$$$\mathrm{1}.\:{Divisibility}\:{by}\:\mathrm{3},\:{i}.{e}.\:\mathrm{3}\mid{P}. \\ $$$${If}\:{any}\:{two}\:{of}\:{the}\:{four}\:{numbers}…
Question Number 19735 by Tinkutara last updated on 15/Aug/17 $$\mathrm{If}\:{z}\:=\:\lambda\:+\:\mathrm{3}\:+\:{i}\sqrt{\mathrm{5}\:−\:\lambda^{\mathrm{2}} },\:\mathrm{then}\:\mathrm{the}\:\mathrm{locus} \\ $$$$\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a} \\ $$ Answered by ajfour last updated on 15/Aug/17 $$\mathrm{if}\:\mathrm{we}\:\mathrm{let}\:\mathrm{z}=\mathrm{x}+\mathrm{iy} \\ $$$$\left(\mathrm{x}−\lambda−\mathrm{3}\right)+\mathrm{i}\left(\mathrm{y}−\sqrt{\mathrm{5}−\lambda^{\mathrm{2}}…