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Question Number 114990 Answers: 0 Comments: 0

Question Number 114988 Answers: 2 Comments: 0

Prove tan^2 x=sin^2 x+sec^2 x

$$\mathrm{Prove}\:\mathrm{tan}^{\mathrm{2}} {x}=\mathrm{sin}^{\mathrm{2}} {x}+\mathrm{sec}^{\mathrm{2}} {x} \\ $$

Question Number 114981 Answers: 2 Comments: 2

Without L′Hopital (1)lim_(x→1) ((x(x+(1/x))^5 −32)/(x−1)) =? (2) lim_(x→∞) (√(2x+(√(2x+(√(2x+(√(2x+(√(...)))))))))) −(√(2x)) = ?

$${Without}\:{L}'{Hopital} \\ $$$$\:\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{{x}\left({x}+\frac{\mathrm{1}}{{x}}\right)^{\mathrm{5}} −\mathrm{32}}{{x}−\mathrm{1}}\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{\mathrm{2}{x}+\sqrt{...}}}}}\:−\sqrt{\mathrm{2}{x}}\:=\:? \\ $$

Question Number 114980 Answers: 1 Comments: 0

Question Number 114978 Answers: 2 Comments: 0

Solve x^2 dy + y(x+y)dx=0

$$ \\ $$$$\:\:{Solve}\: \\ $$$$\:{x}^{\mathrm{2}} {dy}\:+\:{y}\left({x}+{y}\right){dx}=\mathrm{0} \\ $$

Question Number 114975 Answers: 1 Comments: 0

.... nice mathematics.... show that :: Φ = ∫_0 ^( 1) li_2 (x)dx = (π^2 /6) −1 ✓ m.n.july.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:{nice}\:\:{mathematics}....\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{show}\:\:{that}\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {li}_{\mathrm{2}} \left({x}\right){dx}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970} \\ $$$$ \\ $$

Question Number 114967 Answers: 0 Comments: 1

Question Number 114965 Answers: 1 Comments: 0

find all irational numbers x such that x 2+20x+20 and x^(3 ) −2020x+1 both is a rasional number.

$${find}\:{all}\:{irational}\:{numbers}\:{x}\:{such}\:{that}\:{x} \\ $$$$\mathrm{2}+\mathrm{20}{x}+\mathrm{20}\:{and}\:{x}^{\mathrm{3}\:} −\mathrm{2020}{x}+\mathrm{1}\:\:\:{both}\:{is}\:{a} \\ $$$${rasional}\:{number}. \\ $$

Question Number 114996 Answers: 2 Comments: 1

...nice mathematics... prove that::: i:: Σ_(n=1) ^∞ (1/(sinh^2 (πn))) =(1/6) −(1/(2π)) ✓ ii:: Σ_(n=1) ^∞ (n/(e^(2πn) −1))=(1/(24)) −(1/(8π)) ✓✓ iii::Σ_(n=1) ^∞ (1/( nsinh(πn))) =(π/(12))−((ln(2))/4) ✓✓✓ .... M..n..july..1970 ....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{mathematics}...\: \\ $$$$\:\:\:\:{prove}\:{that}::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{i}::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{sinh}^{\mathrm{2}} \left(\pi{n}\right)}\:=\frac{\mathrm{1}}{\mathrm{6}}\:−\frac{\mathrm{1}}{\mathrm{2}\pi}\:\:\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{ii}::\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{8}\pi}\:\:\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{iii}::\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\:{nsinh}\left(\pi{n}\right)}\:=\frac{\pi}{\mathrm{12}}−\frac{{ln}\left(\mathrm{2}\right)}{\mathrm{4}}\:\checkmark\checkmark\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:\:\:\mathscr{M}..{n}..{july}..\mathrm{1970}\:.... \\ $$$$ \\ $$

Question Number 114959 Answers: 2 Comments: 0

long time question proposed by math abdo ∫_0 ^∞ ((lnx)/(1+x^2 +x^4 ))dx

$${long}\:{time}\:{question}\:{proposed}\:{by} \\ $$$${math}\:{abdo} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}{x}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }{dx} \\ $$$$ \\ $$

Question Number 114956 Answers: 0 Comments: 5

Question Number 114950 Answers: 0 Comments: 0

note the tringle ABC is not isosceles with the elevtions of AA1,BB1, and CC1.suppose BA amd CA respectively point at BB1 and CC1 so that A1BA is pependiculer to BB1 and A1CA perpendiculer to CC1.the read and BC lines intersect at the TA point. define in the same way TB and TC poits. prove that TA,TB,and TC are collinear.

$${note}\:{the}\:{tringle}\:{ABC}\:\:{is}\:{not}\:{isosceles}\:{with}\: \\ $$$${the}\:{elevtions}\:{of}\:{AA}\mathrm{1},{BB}\mathrm{1},\:{and}\:{CC}\mathrm{1}.{suppose} \\ $$$${BA}\:\:{amd}\:{CA}\:{respectively}\:{point}\:{at}\:{BB}\mathrm{1}\:{and} \\ $$$${CC}\mathrm{1}\:{so}\:{that}\:{A}\mathrm{1}{BA}\:{is}\:{pependiculer}\:{to}\:{BB}\mathrm{1} \\ $$$${and}\:{A}\mathrm{1}{CA}\:{perpendiculer}\:{to}\:{CC}\mathrm{1}.{the}\:{read}\: \\ $$$${and}\:{BC}\:{lines}\:{intersect}\:{at}\:{the}\:{TA}\:{point}. \\ $$$${define}\:{in}\:{the}\:{same}\:{way}\:\:{TB}\:{and}\:{TC}\:{poits}. \\ $$$${prove}\:{that}\:{TA},{TB},{and}\:{TC}\:{are}\:{collinear}. \\ $$

Question Number 114945 Answers: 1 Comments: 0

Question Number 114943 Answers: 1 Comments: 0

If ^m C_1 =^n C_2 , express m in terms of n.

$$\mathrm{If}\:\:^{{m}} {C}_{\mathrm{1}} =\:^{{n}} {C}_{\mathrm{2}} \:, \\ $$$$\mathrm{express}\:{m}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}. \\ $$

Question Number 114971 Answers: 0 Comments: 2

Question Number 114933 Answers: 2 Comments: 0

find the value of ∫_0 ^∞ ((cos(2x))/(x^4 +x^2 +1))dx

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\mathrm{x}^{\mathrm{4}} \:+\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 114922 Answers: 2 Comments: 0

find minimum value of function y=(√((x+6)^2 +25)) +(√((x−6)^2 +121))

$${find}\:{minimum}\:{value}\:{of}\:{function} \\ $$$${y}=\sqrt{\left({x}+\mathrm{6}\right)^{\mathrm{2}} +\mathrm{25}}\:+\sqrt{\left({x}−\mathrm{6}\right)^{\mathrm{2}} +\mathrm{121}} \\ $$

Question Number 114919 Answers: 0 Comments: 0

Question Number 114912 Answers: 0 Comments: 0

Question Number 114917 Answers: 0 Comments: 3

Question Number 114906 Answers: 3 Comments: 1

Question Number 114880 Answers: 1 Comments: 1

∫ln (sin (x))dx=?

$$\int\mathrm{ln}\:\left(\mathrm{sin}\:\left({x}\right)\right){dx}=? \\ $$

Question Number 114879 Answers: 1 Comments: 0

If the product of the matrices (((1 1)),((0 1)) ) (((1 2)),((0 1)) ) (((1 3)),((0 1)) )... (((1 k)),((0 1)) )= (((1 378)),((0 1)) ) then k =

$${If}\:{the}\:{product}\:{of}\:{the}\:{matrices}\: \\ $$$$\begin{pmatrix}{\mathrm{1}\:\:\:\mathrm{1}}\\{\mathrm{0}\:\:\:\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{2}}\\{\mathrm{0}\:\:\:\:\mathrm{1}}\end{pmatrix}\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\mathrm{3}}\\{\mathrm{0}\:\:\:\:\:\mathrm{1}}\end{pmatrix}...\begin{pmatrix}{\mathrm{1}\:\:\:\:\:{k}}\\{\mathrm{0}\:\:\:\:\:\mathrm{1}}\end{pmatrix}=\begin{pmatrix}{\mathrm{1}\:\:\:\:\mathrm{378}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$${then}\:{k}\:=\: \\ $$

Question Number 114878 Answers: 2 Comments: 0

Given f(x) = ∫_0 ^x (dt/( (√(1+t^3 )))) and g(x) be the inverse function of f(x), then g ′′(x)=λg^2 (x). then the value of λ =

$${Given}\:{f}\left({x}\right)\:=\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{3}} }}\:{and}\:{g}\left({x}\right)\:{be}\:{the} \\ $$$${inverse}\:{function}\:{of}\:{f}\left({x}\right),\:{then}\:{g}\:''\left({x}\right)=\lambda{g}^{\mathrm{2}} \left({x}\right). \\ $$$${then}\:{the}\:{value}\:{of}\:\lambda\:= \\ $$

Question Number 114876 Answers: 2 Comments: 1

Question Number 114875 Answers: 2 Comments: 0

A man sent 7 letters to his 7 friend . the letters are kept in addressed envelopes at random. the probability that 3 friends receive correct letters and 4 letters go to wrong destination is _ (old question unanswered)

$${A}\:{man}\:{sent}\:\mathrm{7}\:{letters}\:{to}\:{his}\:\mathrm{7}\:{friend}\:. \\ $$$${the}\:{letters}\:{are}\:{kept}\:{in}\:{addressed}\:{envelopes} \\ $$$${at}\:{random}.\:{the}\:{probability}\:{that}\:\mathrm{3}\:{friends} \\ $$$${receive}\:{correct}\:{letters}\:{and}\:\mathrm{4}\:{letters}\:{go} \\ $$$${to}\:{wrong}\:{destination}\:{is}\:\_ \\ $$$$ \\ $$$$\left({old}\:{question}\:{unanswered}\right) \\ $$

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