Question Number 131552 by liberty last updated on 06/Feb/21 $$\int\:\frac{\mathrm{x}\:\mathrm{dx}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }\:? \\ $$ Answered by EDWIN88 last updated on 06/Feb/21 $$\Leftrightarrow\:\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}\:=\:\frac{\mathrm{2}}{\mathrm{sin}\:\mathrm{2x}} \\ $$$$\Leftrightarrow\:\frac{\mathrm{1}}{\left(\mathrm{cot}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}\right)^{\mathrm{2}} }=\frac{\mathrm{sin}\:^{\mathrm{2}} \mathrm{2x}}{\mathrm{4}}…
Question Number 66019 by Rio Michael last updated on 07/Aug/19 $$\underset{{x}\rightarrow\infty} {{lim}}\:\left(\mathrm{1}\:+\:\frac{\mathrm{2}}{{x}}\right)^{{x}} \:= \\ $$ Commented by mathmax by abdo last updated on 07/Aug/19 $${let}\:{f}\left({x}\right)=\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{{x}}…
Question Number 482 by prakash jain last updated on 12/Jan/15 $$\mathrm{The}\:\mathrm{number}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{certain}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{0}{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end},\:\mathrm{what}\:\mathrm{the}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero} \\ $$$$\mathrm{digit}. \\ $$$$\mathrm{1000}!=…\mathrm{d}_{\mathrm{1}} \mathrm{d}_{\mathrm{2}} \mathrm{d}_{\mathrm{3}} \mathrm{D00000}… \\ $$$$\mathrm{where}\:\mathrm{d}_{\mathrm{1}} ,\mathrm{d}_{\mathrm{2}} ,\mathrm{d}_{\mathrm{3}} ,\mathrm{D}\:\mathrm{are}\:\mathrm{digits}.…
Question Number 66016 by Rio Michael last updated on 07/Aug/19 $${Evaluate}\:\:\: \\ $$$${a}.\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\left({lnx}\right)^{\mathrm{2}} {dx} \\ $$$${b}.\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{3}} {xdx} \\ $$ Commented…
Question Number 481 by 123456 last updated on 12/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:\left\{{x}_{{n}} \right\}\:{is}\:{a}\:{no}\:{limited}\:{sequence} \\ $$$${then} \\ $$$${exist}\:{a}\:{sub}−{sequence}\:\left\{{x}_{{nk}} \right\}\:{that} \\ $$$$\underset{{n}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}_{{nk}} }=\mathrm{0} \\ $$ Commented…
Question Number 131554 by Salman_Abir last updated on 06/Feb/21 Commented by EDWIN88 last updated on 06/Feb/21 $$\mathrm{A}−\mathrm{B}=\:\begin{bmatrix}{−\mathrm{2}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{6}}\end{bmatrix} \\ $$$$\mathrm{A}×\mathrm{B}^{\mathrm{T}} \:=\:\begin{bmatrix}{\mathrm{2}\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\mathrm{4}}\\{\mathrm{1}\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\mathrm{7}}\end{bmatrix}\begin{bmatrix}{\mathrm{4}\:\:\:\:\:\:−\mathrm{1}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{4}}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\mathrm{1}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\begin{bmatrix}{\mathrm{8}+\mathrm{6}+\mathrm{12}\:\:\:\:\:\:\:−\mathrm{2}+\mathrm{12}+\mathrm{4}}\\{\mathrm{4}+\mathrm{10}+\mathrm{21}\:\:\:\:\:\:−\mathrm{1}+\mathrm{20}+\mathrm{7}}\end{bmatrix} \\ $$ Terms…
Question Number 66017 by Rio Michael last updated on 07/Aug/19 $${show}\:{that}\:{the}\:{equation}\:{xe}^{{x}} =\mathrm{1}\:{has}\:{a}\:{root}\:{between}\:\mathrm{0}.\mathrm{5}\:{and}\:\mathrm{0}.\mathrm{6}\:{starting} \\ $$$${with}\:\mathrm{0}.\mathrm{55}\:{as}\:{a}\:{first}\:{approximate}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 480 by 123456 last updated on 11/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${for}\:{s}\in\left\{\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{\mathrm{1}}{{s}^{{i}} }−\frac{\left(−\mathrm{1}\right)^{{i}} }{{i}^{{s}} }\right]\leqslant\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{s}+\mathrm{1}}{{si}^{{s}} } \\ $$ Commented…
Question Number 131549 by liberty last updated on 06/Feb/21 $$\mathrm{slowly}\:\mathrm{integral}\: \\ $$$$\int\:\frac{\mathrm{sec}\:^{\mathrm{4}} \mathrm{x}}{\:\sqrt{\mathrm{tan}\:^{\mathrm{3}} \mathrm{x}}}\:\mathrm{dx}\:=? \\ $$ Answered by rs4089 last updated on 06/Feb/21 $$\frac{\mathrm{2}{tan}^{\mathrm{2}} {x}−\mathrm{6}}{\mathrm{3}\sqrt{{tanx}}}+{c}\:\:\:\:\left\{{c}\:{is}\:{a}\:{constant}\right\}…
Question Number 131548 by liberty last updated on 06/Feb/21 $$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{n}}\:\underset{\mathrm{0}} {\overset{\:\:\frac{\pi}{\mathrm{2}}} {\int}}\:\mathrm{cos}\:^{\mathrm{2n}+\mathrm{1}} \left(\theta\right)\:\mathrm{d}\theta\:=? \\ $$ Answered by rs4089 last updated on 06/Feb/21 $$\frac{\pi\sqrt{\pi}}{\mathrm{2}} \\…