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Question Number 106908 Answers: 1 Comments: 0

Question Number 106907 Answers: 1 Comments: 0

Question Number 106906 Answers: 2 Comments: 0

Question Number 106899 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((tsin(2t))/(t^2 +4)) dt

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{tsin}\left(\mathrm{2t}\right)}{\mathrm{t}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{dt} \\ $$

Question Number 106888 Answers: 2 Comments: 0

@bemath@ given { ((f(x)=log _2 (sin x)+log _2 (cos x))),((g(x)=log _2 (cos 2x)+log _2 (cos 4x))) :} find f((π/(48)))+g((π/(48))) .

$$@\mathrm{bemath}@ \\ $$$$\mathfrak{g}\mathrm{iven}\:\begin{cases}{\mathrm{f}\left(\mathrm{x}\right)=\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{sin}\:\mathrm{x}\right)+\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{cos}\:\mathrm{x}\right)}\\{\mathrm{g}\left(\mathrm{x}\right)=\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{cos}\:\mathrm{2x}\right)+\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{cos}\:\mathrm{4x}\right)}\end{cases} \\ $$$$\mathrm{find}\:\mathrm{f}\left(\frac{\pi}{\mathrm{48}}\right)+\mathrm{g}\left(\frac{\pi}{\mathrm{48}}\right)\:. \\ $$

Question Number 106883 Answers: 2 Comments: 0

◊JS⧫ lim_(x→−1) (((√(x^2 −1))+1−(√(−x)))/(√(x+1))) ?

$$\:\:\:\:\:\:\:\lozenge\mathrm{JS}\blacklozenge \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}+\mathrm{1}−\sqrt{−\mathrm{x}}}{\sqrt{\mathrm{x}+\mathrm{1}}}\:?\: \\ $$

Question Number 106880 Answers: 3 Comments: 2

^(@bemath@) What is m so the roots of x^4 −(m+2)x^2 +9=0 are in AP

$$\:\:\:\:\:\:\:\:\:\:\overset{@\mathrm{bemath}@} {\:} \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{m}\:\mathrm{so}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{x}^{\mathrm{4}} \:−\left(\mathrm{m}+\mathrm{2}\right)\mathrm{x}^{\mathrm{2}} +\mathrm{9}=\mathrm{0} \\ $$$$\mathrm{are}\:\mathrm{in}\:\mathrm{AP} \\ $$

Question Number 106869 Answers: 1 Comments: 0

@bemath@ lim_(x→π) [((1/(2x−2π))) ∫_π ^x ((cos 2t dt)/(1−cos 3t)) ]=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\left[\left(\frac{\mathrm{1}}{\mathrm{2x}−\mathrm{2}\pi}\right)\:\underset{\pi} {\overset{\mathrm{x}} {\int}}\:\frac{\mathrm{cos}\:\mathrm{2t}\:\mathrm{dt}}{\mathrm{1}−\mathrm{cos}\:\mathrm{3t}}\:\right]=? \\ $$

Question Number 106867 Answers: 2 Comments: 3

∀n∈(0, 1)∀x∈R : f(x) = (n/(n−1))x+n The given function gives a linear line that goes through points (0, n) and (1−n, 0). The function changes as n changes. What area is beneath the shape made as n goes from 0→1?

$$\forall{n}\in\left(\mathrm{0},\:\mathrm{1}\right)\forall{x}\in\mathbb{R}\::\:{f}\left({x}\right)\:=\:\frac{{n}}{{n}−\mathrm{1}}{x}+{n} \\ $$$$\mathrm{The}\:\mathrm{given}\:\mathrm{function}\:\mathrm{gives}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{line}\:\mathrm{that} \\ $$$$\mathrm{goes}\:\mathrm{through}\:\mathrm{points}\:\left(\mathrm{0},\:{n}\right)\:\mathrm{and}\:\left(\mathrm{1}−{n},\:\mathrm{0}\right). \\ $$$$\mathrm{The}\:\mathrm{function}\:\mathrm{changes}\:\mathrm{as}\:{n}\:\mathrm{changes}. \\ $$$$\mathrm{What}\:\mathrm{area}\:\mathrm{is}\:\mathrm{beneath}\:\mathrm{the}\:\mathrm{shape}\:\mathrm{made}\:\mathrm{as} \\ $$$${n}\:\mathrm{goes}\:\mathrm{from}\:\mathrm{0}\rightarrow\mathrm{1}? \\ $$

Question Number 106860 Answers: 0 Comments: 3

1−1+(5/9)−(7/(27))+(9/(81))−((11)/(243))+.....

$$\mathrm{1}−\mathrm{1}+\frac{\mathrm{5}}{\mathrm{9}}−\frac{\mathrm{7}}{\mathrm{27}}+\frac{\mathrm{9}}{\mathrm{81}}−\frac{\mathrm{11}}{\mathrm{243}}+..... \\ $$

Question Number 106855 Answers: 0 Comments: 0

Question Number 106847 Answers: 3 Comments: 0

Given f(x)=((3(√3))/(sinx))+(1/(cosx)) show that f ′(x)=cosx(((tan^3 x−3(√3)))/(sin^2 x))

$${Given}\:{f}\left({x}\right)=\frac{\mathrm{3}\sqrt{\mathrm{3}}}{{sinx}}+\frac{\mathrm{1}}{{cosx}} \\ $$$${show}\:{that}\:{f}\:'\left({x}\right)={cosx}\frac{\left({tan}^{\mathrm{3}} {x}−\mathrm{3}\sqrt{\mathrm{3}}\right)}{{sin}^{\mathrm{2}} {x}} \\ $$

Question Number 106844 Answers: 0 Comments: 2

Please any geometry proof on shortest distance between two places (Topic longitude and latitude).

$$\mathrm{Please}\:\mathrm{any}\:\mathrm{geometry}\:\mathrm{proof}\:\mathrm{on}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{two} \\ $$$$\mathrm{places}\:\:\:\left(\mathrm{Topic}\:\mathrm{longitude}\:\mathrm{and}\:\mathrm{latitude}\right). \\ $$

Question Number 106842 Answers: 2 Comments: 5

○bobhans○ lim_(x→0) ((((1+sin 4x))^(1/3) −cos 2x)/(x tan x)) ?

$$\:\:\:\:\:\circ\mathrm{bobhans}\circ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{1}+\mathrm{sin}\:\mathrm{4x}}\:−\mathrm{cos}\:\mathrm{2x}}{\mathrm{x}\:\mathrm{tan}\:\mathrm{x}}\:? \\ $$

Question Number 106830 Answers: 1 Comments: 7

∫(( dx)/((√(x^3 )) ^3 (√(1 +^4 (√x^3 ))))) = ?

$$\int\frac{\:{dx}}{\sqrt{{x}^{\mathrm{3}} \:}\:\:^{\mathrm{3}} \sqrt{\mathrm{1}\:+\:^{\mathrm{4}} \sqrt{{x}^{\mathrm{3}} }}}\:=\:? \\ $$

Question Number 106828 Answers: 0 Comments: 2

∫ (1/(xdx)) is that true!

$$\:\int\:\frac{\mathrm{1}}{{xdx}}\:{is}\:{that}\:{true}! \\ $$

Question Number 106825 Answers: 5 Comments: 0

lim_(x→0) ((sin5x − tan5x)/x^3 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\mathrm{5}{x}\:−\:{tan}\mathrm{5}{x}}{{x}^{\mathrm{3}} } \\ $$

Question Number 106816 Answers: 2 Comments: 1

prove by mathematical induction (1) n^2 ≤ 2^n ; n ≥ 4 (2) (n+1)^2 < 2n^2 ; n ≥ 3 (3) 2^n −3 ≥ 2^(n−2) ; n ≥ 5 @bemath@

$$\mathrm{prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{n}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{\mathrm{n}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{2}} \:<\:\mathrm{2n}^{\mathrm{2}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{3} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{2}^{\mathrm{n}} −\mathrm{3}\:\geqslant\:\mathrm{2}^{\mathrm{n}−\mathrm{2}} \:;\:\mathrm{n}\:\geqslant\:\mathrm{5} \\ $$$$\:\:\:\:\:\:@\mathrm{bemath}@ \\ $$

Question Number 106815 Answers: 1 Comments: 0

please help me to show that the equation X^n +aX+c=0 can not have more than 3 reals solutions

$$\:\:\:\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{to}}\:\boldsymbol{{show}} \\ $$$$\boldsymbol{{that}}\:\:\boldsymbol{{the}}\:\boldsymbol{{equation}}\: \\ $$$$\:\boldsymbol{{X}}^{\boldsymbol{{n}}} +\boldsymbol{{aX}}+\boldsymbol{{c}}=\mathrm{0}\:\boldsymbol{{can}}\:\boldsymbol{{not}}\:\boldsymbol{{have}} \\ $$$$\boldsymbol{{more}}\:\boldsymbol{{than}}\:\mathrm{3}\:\boldsymbol{{reals}}\:\boldsymbol{{solutions}} \\ $$$$ \\ $$

Question Number 106810 Answers: 3 Comments: 0

lim_(x→0) ((tanx−sinx)/(sinx(cos2x−cosx)))=???

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{tanx}−{sinx}}{{sinx}\left({cos}\mathrm{2}{x}−{cosx}\right)}=??? \\ $$

Question Number 106809 Answers: 2 Comments: 0

∫e^(2x) sine^x dx=??

$$\int{e}^{\mathrm{2}{x}} {sine}^{{x}} {dx}=?? \\ $$

Question Number 106808 Answers: 0 Comments: 0

∫tan(lnx)dx=???

$$\int{tan}\left({lnx}\right){dx}=??? \\ $$

Question Number 106807 Answers: 1 Comments: 0

∫sin(ln3x)dx=???

$$\int{sin}\left({ln}\mathrm{3}{x}\right){dx}=??? \\ $$

Question Number 106794 Answers: 3 Comments: 0

repost old question unanswer Given → { ((((4x^2 )/(1+4x^2 )) = y)),((((4y^2 )/(1+4y^2 )) = z)),((((4z^2 )/(1+4z^2 )) = x)) :}

$$\mathrm{repost}\:\mathrm{old}\:\mathrm{question}\:\mathrm{unanswer} \\ $$$$\mathcal{G}\mathrm{iven}\:\rightarrow\begin{cases}{\frac{\mathrm{4x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4x}^{\mathrm{2}} }\:=\:\mathrm{y}}\\{\frac{\mathrm{4y}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4y}^{\mathrm{2}} }\:=\:\mathrm{z}}\\{\frac{\mathrm{4z}^{\mathrm{2}} }{\mathrm{1}+\mathrm{4z}^{\mathrm{2}} }\:=\:\mathrm{x}}\end{cases} \\ $$

Question Number 106792 Answers: 1 Comments: 0

≻bobhans≺ From a batch containing 6 boys and 4 girls a group of 4 students is tobe selected . How many group formations will have exactly 2 girls?

$$\:\:\:\:\:\:\succ\mathrm{bobhans}\prec \\ $$$$\mathrm{From}\:\mathrm{a}\:\mathrm{batch}\:\mathrm{containing}\:\mathrm{6}\:\mathrm{boys}\:\mathrm{and}\:\mathrm{4}\:\mathrm{girls} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\mathrm{4}\:\mathrm{students}\:\mathrm{is}\:\mathrm{tobe}\:\mathrm{selected}\:. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{group}\:\mathrm{formations}\:\mathrm{will}\:\mathrm{have} \\ $$$$\mathrm{exactly}\:\mathrm{2}\:\mathrm{girls}? \\ $$

Question Number 106787 Answers: 2 Comments: 0

^(@bemath@) y = (cos 2x)^(sin x) , find (dy/dx) ?

$$\:\:\:\:\:\:\overset{@\mathrm{bemath}@} {\:} \\ $$$$\:\:\mathrm{y}\:=\:\left(\mathrm{cos}\:\mathrm{2x}\right)^{\mathrm{sin}\:\mathrm{x}} \:,\:\mathrm{find}\:\frac{\mathrm{dy}}{\mathrm{dx}}\:? \\ $$

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