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Question Number 136604    Answers: 2   Comments: 0

What is the equation of a tangent to a circle whose equation is x^2+y^2−2x−4y=1 at point (1+√5,3)

$$ \\ $$What is the equation of a tangent to a circle whose equation is x^2+y^2−2x−4y=1 at point (1+√5,3)

Question Number 136598    Answers: 1   Comments: 0

Question Number 136597    Answers: 1   Comments: 0

Question Number 136596    Answers: 1   Comments: 0

Question Number 136636    Answers: 0   Comments: 1

b=a+c y=x+c x=a+z y=b+z x=?,y=?,z=?

$${b}={a}+{c} \\ $$$${y}={x}+{c} \\ $$$${x}={a}+{z} \\ $$$${y}={b}+{z} \\ $$$${x}=?,{y}=?,{z}=? \\ $$

Question Number 136635    Answers: 0   Comments: 0

Question Number 136582    Answers: 1   Comments: 0

Use De′Moivre′s theorem to prove that Σ_(r=0) ^∞ cos rx = (1/2) and find a value(or expression) for Σ_(r=0) ^∞ sin rx assume that this two series were convergent.

$$\mathrm{Use}\:\mathrm{De}'\mathrm{Moivre}'\mathrm{s}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{cos}\:{rx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{find}\:\mathrm{a}\:\mathrm{value}\left(\mathrm{or}\:\mathrm{expression}\right)\:\mathrm{for}\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\mathrm{sin}\:{rx} \\ $$$$\mathrm{assume}\:\mathrm{that}\:\mathrm{this}\:\mathrm{two}\:\mathrm{series}\:\mathrm{were}\:\mathrm{convergent}. \\ $$

Question Number 136581    Answers: 1   Comments: 0

.......advanced calculus.... pove that:: :::Σ_(n=0) ^∞ {(((−1)^n )/2^(2n) ) ((( 2n)),(( n)) ) cos(nx)}=((cos((x/4)))/( (√(2cos((x/2))))))

$$\:\:\:\:\:\:\:\:.......{advanced}\:\:\:{calculus}.... \\ $$$$\:\:\:{pove}\:{that}:: \\ $$$$:::\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{2}^{\mathrm{2}{n}} }\begin{pmatrix}{\:\mathrm{2}{n}}\\{\:\:{n}}\end{pmatrix}\:{cos}\left({nx}\right)\right\}=\frac{{cos}\left(\frac{{x}}{\mathrm{4}}\right)}{\:\sqrt{\mathrm{2}{cos}\left(\frac{{x}}{\mathrm{2}}\right)}} \\ $$

Question Number 136572    Answers: 3   Comments: 1

....advanced calculus.... 𝛗=∫_0 ^( (π/2)) x.(tan(x))^(1/2) dx=??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:....{advanced}\:\:\:\:{calculus}.... \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {x}.\left({tan}\left({x}\right)\right)^{\frac{\mathrm{1}}{\mathrm{2}}} {dx}=?? \\ $$$$ \\ $$

Question Number 136570    Answers: 1   Comments: 0

.......nice ..... calculus..... Ω=Σ_(n=0) ^∞ cos^n (x).cos(nx)=? solution:::: Ω=(1/2)Σ_(n=0) ^∞ cos^(n−1) (x){cos(x−nx)+cos(x+nx) ∴ 2Ω=Σ_(n=0) ^∞ cos^(n−1) (x).cos(n−1)x +Σ_(n=0) ^∞ cos^(n−1) (x).cos(n+1)x =1+Σ_(n=1) ^∞ cos^(n−1) (x).cos(n−1)x+(1/(cos^2 (x)))Σ_(n=0) ^∞ cos^(n+1) (x).cos(n+1)x =1+Ω+(1/(cos^2 (x)))(Ω−1) Ω(1−(1/(cos^2 (x))))=1−(1/(cos^2 (x))) ∴ Ω=1 .................

$$\:\:\:\:\:\:\:\:\:.......{nice}\:\:.....\:\:\:{calculus}..... \\ $$$$\:\:\:\:\Omega=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{cos}^{{n}} \left({x}\right).{cos}\left({nx}\right)=? \\ $$$$\:\:\:{solution}:::: \\ $$$$\:\:\:\:\Omega=\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{cos}^{{n}−\mathrm{1}} \left({x}\right)\left\{{cos}\left({x}−{nx}\right)+{cos}\left({x}+{nx}\right)\right. \\ $$$$\:\:\therefore\:\mathrm{2}\Omega=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{cos}^{{n}−\mathrm{1}} \left({x}\right).{cos}\left({n}−\mathrm{1}\right){x} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{cos}^{{n}−\mathrm{1}} \left({x}\right).{cos}\left({n}+\mathrm{1}\right){x} \\ $$$$\:\:\:\:\:\:\:\:\:=\mathrm{1}+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{cos}^{{n}−\mathrm{1}} \left({x}\right).{cos}\left({n}−\mathrm{1}\right){x}+\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{cos}^{{n}+\mathrm{1}} \left({x}\right).{cos}\left({n}+\mathrm{1}\right){x} \\ $$$$\:\:\:\:\:\:\:\:\:=\mathrm{1}+\Omega+\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}\left(\Omega−\mathrm{1}\right) \\ $$$$\:\:\:\Omega\left(\mathrm{1}−\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}\right)=\mathrm{1}−\frac{\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)} \\ $$$$\:\:\:\:\:\therefore\:\:\:\:\:\:\:\:\:\Omega=\mathrm{1}\:................. \\ $$$$\:\: \\ $$

Question Number 136559    Answers: 4   Comments: 1

show that Σ_(n=2) ^∞ ((n−4)/(n^2 −2n+1)) is converge or diverge by ussing any test

$${show}\:{that}\:\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{{n}−\mathrm{4}}{{n}^{\mathrm{2}} −\mathrm{2}{n}+\mathrm{1}}\:{is}\:{converge}\:{or}\:{diverge}\: \\ $$$$ \\ $$$${by}\:{ussing}\:{any}\:{test} \\ $$

Question Number 136555    Answers: 3   Comments: 0

....nice ...... calculus.... prove that 𝛗=∫_0 ^( ∞) ((ln(1+x^2 ))/((1+x^2 )^2 ))dx=(π/2)ln(2)−(π/4) ::::::::

$$\:\:\:\:\:\:\:\:\:....{nice}\:\:\:\:......\:\:\:{calculus}.... \\ $$$$\:\:\:{prove}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}}{ln}\left(\mathrm{2}\right)−\frac{\pi}{\mathrm{4}} \\ $$$$\:\:\:\:\::::::::: \\ $$

Question Number 136551    Answers: 0   Comments: 2

Google released an update to android webview which is causing multiple android app to crash. Please search internet for workaround for your device. This problem is not caused by Tinkutara Equation Editor. Please also note no android app can cause other apps to crash. Apps released by google and your device manufacturer are system wide apps and those can cause system wide problems.

$$\mathrm{Google}\:\mathrm{released}\:\mathrm{an}\:\mathrm{update}\:\mathrm{to}\:\mathrm{android} \\ $$$$\mathrm{webview}\:\mathrm{which}\:\mathrm{is}\:\mathrm{causing}\:\mathrm{multiple} \\ $$$$\mathrm{android}\:\mathrm{app}\:\mathrm{to}\:\mathrm{crash}.\:\mathrm{Please} \\ $$$$\mathrm{search}\:\mathrm{internet}\:\mathrm{for}\:\mathrm{workaround} \\ $$$$\mathrm{for}\:\mathrm{your}\:\mathrm{device}. \\ $$$$\boldsymbol{\mathrm{This}}\:\boldsymbol{\mathrm{problem}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{not}}\:\boldsymbol{\mathrm{caused}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{Tinkutara}} \\ $$$$\boldsymbol{\mathrm{Equation}}\:\boldsymbol{\mathrm{Editor}}. \\ $$$$\mathrm{Please}\:\mathrm{also}\:\mathrm{note}\:\mathrm{no}\:\mathrm{android}\:\mathrm{app} \\ $$$$\mathrm{can}\:\mathrm{cause}\:\mathrm{other}\:\mathrm{apps}\:\mathrm{to}\:\mathrm{crash}. \\ $$$$\mathrm{Apps}\:\mathrm{released}\:\mathrm{by}\:\mathrm{google}\:\mathrm{and}\:\mathrm{your} \\ $$$$\mathrm{device}\:\mathrm{manufacturer}\:\mathrm{are}\:\mathrm{system} \\ $$$$\mathrm{wide}\:\mathrm{apps}\:\mathrm{and}\:\mathrm{those}\:\mathrm{can}\:\mathrm{cause} \\ $$$$\mathrm{system}\:\mathrm{wide}\:\mathrm{problems}. \\ $$

Question Number 136541    Answers: 0   Comments: 13

to Tinku Tara sir: since update to version 2.262 i′m encountering a big problem: after some time of use the app crashes, then i can′t start the app any more, that means after displaying the welcome screen the app crashes. besides some other apps, e.g. google, crash too. i have uninstalled and reinstalled the app many times. i expect, after i have sent this message, if i can send, the app will crash. when i delete the app, other apps run properly, therefore i think it has to do with the new version, because i never had this problem before. my device is Huawei P20. can you please provide a link to download the old version? i′ll do some experiments to see which version causes the problem.

$${to}\:{Tinku}\:\:{Tara}\:{sir}: \\ $$$${since}\:{update}\:{to}\:{version}\:\mathrm{2}.\mathrm{262}\:{i}'{m} \\ $$$${encountering}\:{a}\:{big}\:{problem}: \\ $$$${after}\:{some}\:{time}\:{of}\:{use}\:{the}\:{app}\: \\ $$$${crashes},\:{then}\:{i}\:{can}'{t}\:{start}\:{the}\:{app} \\ $$$${any}\:{more},\:{that}\:{means}\:{after}\:{displaying} \\ $$$${the}\:{welcome}\:{screen}\:{the}\:{app}\:{crashes}. \\ $$$${besides}\:{some}\:{other}\:{apps},\:{e}.{g}.\:{google}, \\ $$$${crash}\:{too}. \\ $$$${i}\:{have}\:{uninstalled}\:{and}\:{reinstalled} \\ $$$${the}\:{app}\:{many}\:{times}. \\ $$$${i}\:{expect},\:{after}\:{i}\:{have}\:{sent}\:{this} \\ $$$${message},\:{if}\:{i}\:{can}\:{send},\:{the}\:{app}\:{will} \\ $$$${crash}. \\ $$$${when}\:{i}\:{delete}\:{the}\:{app},\:{other}\:{apps} \\ $$$${run}\:{properly},\:{therefore}\:{i}\:{think}\:{it}\:{has} \\ $$$${to}\:{do}\:{with}\:{the}\:{new}\:{version},\:{because} \\ $$$${i}\:{never}\:{had}\:{this}\:{problem}\:{before}. \\ $$$${my}\:{device}\:{is}\:{Huawei}\:{P}\mathrm{20}. \\ $$$${can}\:{you}\:{please}\:{provide}\:{a}\:{link}\:{to} \\ $$$${download}\:{the}\:{old}\:{version}?\:{i}'{ll}\:{do} \\ $$$${some}\:{experiments}\:{to}\:{see}\:{which} \\ $$$${version}\:{causes}\:{the}\:{problem}. \\ $$

Question Number 136594    Answers: 0   Comments: 0

lim_0^+ (x^x^x /(x^x −1))

$${li}\underset{\mathrm{0}^{+} } {{m}}\frac{{x}^{{x}^{{x}} } }{{x}^{{x}} −\mathrm{1}} \\ $$

Question Number 136593    Answers: 1   Comments: 0

Question Number 136537    Answers: 1   Comments: 0

Question Number 136536    Answers: 0   Comments: 0

Question Number 136520    Answers: 3   Comments: 1

Question Number 136519    Answers: 1   Comments: 0

1−((1/2))^(2k) +(((1.3)/(2.4)))^(2k) −(((1.3.5)/(2.4.6)))^(2k) +(((1.3.5.7)/(2.4.6.8)))^(2k) −.... Find the general form

$$\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}{k}} +\left(\frac{\mathrm{1}.\mathrm{3}}{\mathrm{2}.\mathrm{4}}\right)^{\mathrm{2}{k}} −\left(\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}}{\mathrm{2}.\mathrm{4}.\mathrm{6}}\right)^{\mathrm{2}{k}} +\left(\frac{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}}{\mathrm{2}.\mathrm{4}.\mathrm{6}.\mathrm{8}}\right)^{\mathrm{2}{k}} −.... \\ $$$${Find}\:{the}\:{general}\:{form} \\ $$

Question Number 136516    Answers: 1   Comments: 0

find all integers (x,y) : 5^x =3^x +2021y

$${find}\:{all}\:{integers}\:\left({x},{y}\right)\::\:\mathrm{5}^{{x}} =\mathrm{3}^{{x}} +\mathrm{2021}{y} \\ $$

Question Number 136514    Answers: 1   Comments: 0

if z=((1−i(√3))/2) find arg(−z)

$${if}\:{z}=\frac{\mathrm{1}−{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:\:{find}\:{arg}\left(−{z}\right) \\ $$

Question Number 136505    Answers: 1   Comments: 0

Question Number 136500    Answers: 2   Comments: 0

Question Number 136494    Answers: 3   Comments: 0

5((√(1−x)) +(√(1+x)) )= 6x + 8(√(1−x^2 ))

$$\mathrm{5}\left(\sqrt{\mathrm{1}−{x}}\:+\sqrt{\mathrm{1}+{x}}\:\right)=\:\mathrm{6}{x}\:+\:\mathrm{8}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\: \\ $$

Question Number 136531    Answers: 0   Comments: 0

L=lim_(x→0) ((2(1+sinx)^(1/(sinx)) −(1+tanx)^(1/(tanx)) −e)/x^2 )

$$\mathrm{L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}\left(\mathrm{1}+\mathrm{sinx}\right)^{\frac{\mathrm{1}}{\mathrm{sinx}}} −\left(\mathrm{1}+\mathrm{tanx}\right)^{\frac{\mathrm{1}}{\mathrm{tanx}}} −\mathrm{e}}{\mathrm{x}^{\mathrm{2}} } \\ $$

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