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AllQuestion and Answers: Page 1180

Question Number 98984 Answers: 0 Comments: 0

Question Number 98983 Answers: 0 Comments: 2

let a,b,c be positive real numbers such that ab+bc+ac=3 prove the inquality ((a(b^2 +c^2 ))/(a^2 +bc))+((b(c^2 +a^2 ))/(b^2 +ac))+((c(b^2 +a^2 ))/(c^2 +ab))≥3

$${let}\:{a},{b},{c}\:{be}\:{positive}\:{real}\:{numbers}\:{such} \\ $$$${that}\:{ab}+{bc}+{ac}=\mathrm{3}\: \\ $$$${prove}\:{the}\:{inquality} \\ $$$$ \\ $$$$\frac{{a}\left({b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right)}{{a}^{\mathrm{2}} +{bc}}+\frac{{b}\left({c}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{b}^{\mathrm{2}} +{ac}}+\frac{{c}\left({b}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{c}^{\mathrm{2}} +{ab}}\geqslant\mathrm{3} \\ $$

Question Number 98968 Answers: 0 Comments: 1

if x^x^x^x^(2020) = 2020. Solve for x

$$\mathrm{if}\:\mathrm{x}^{\mathrm{x}^{\mathrm{x}^{\mathrm{x}^{\mathrm{2020}} } } } =\:\mathrm{2020}.\:\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{x} \\ $$$$ \\ $$

Question Number 98967 Answers: 1 Comments: 0

solve: (((∫_2 ^6 x(√(1+9⌊x⌋^2 ))dx)/(∫_0 ^1 x{(1/x)}⌈(1/x)⌉dx)))(Σ_(n≥1) (−1)^n ((Π_(j=1) ^n ((3/2)−j))/((2n+1)n!)))

$${solve}: \\ $$$$\left(\frac{\int_{\mathrm{2}} ^{\mathrm{6}} {x}\sqrt{\mathrm{1}+\mathrm{9}\lfloor{x}\rfloor^{\mathrm{2}} }{dx}}{\int_{\mathrm{0}} ^{\mathrm{1}} {x}\left\{\frac{\mathrm{1}}{{x}}\right\}\lceil\frac{\mathrm{1}}{{x}}\rceil{dx}}\right)\left(\underset{{n}\geqslant\mathrm{1}} {\sum}\left(−\mathrm{1}\right)^{{n}} \frac{\prod_{{j}=\mathrm{1}} ^{{n}} \left(\frac{\mathrm{3}}{\mathrm{2}}−{j}\right)}{\left(\mathrm{2}{n}+\mathrm{1}\right){n}!}\right) \\ $$

Question Number 98955 Answers: 2 Comments: 2

lim_(x→0) (√(x+(√(x+(√(x+(√(x+∙∙∙))))))))

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\centerdot\centerdot\centerdot}}}} \\ $$

Question Number 98953 Answers: 2 Comments: 0

Given f(x)=sin^2 x find the expansion of f(x) up to the n^(th) term.

$$\mathcal{G}\mathrm{iven}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{sin}^{\mathrm{2}} \mathrm{x}\:\mathrm{find}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right) \\ $$$$\mathrm{up}\:\mathrm{to}\:\mathrm{the}\:\mathrm{n}^{\mathrm{th}} \:\mathrm{term}. \\ $$

Question Number 98952 Answers: 0 Comments: 1

Without using L′Ho^ pital′s rule or Maclaurin′s expansion series, find lim_(x→0) ((((xe^x )/(e^x −1))−1)/x)

$$\mathrm{Without}\:\mathrm{using}\:\mathrm{L}'\mathrm{H}\hat {\mathrm{o}pital}'\mathrm{s}\:\mathrm{rule}\:\mathrm{or}\:\mathrm{Maclaurin}'\mathrm{s}\:\mathrm{expansion} \\ $$$$\mathrm{series},\:\mathrm{find}\:\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{xe}^{\mathrm{x}} }{\mathrm{e}^{\mathrm{x}} −\mathrm{1}}−\mathrm{1}}{\mathrm{x}} \\ $$

Question Number 98951 Answers: 0 Comments: 1

∫tan^(1/5) x.cotx.secxdx

$$\int{tan}^{\frac{\mathrm{1}}{\mathrm{5}}} {x}.{cotx}.{secxdx} \\ $$

Question Number 98945 Answers: 0 Comments: 0

calculate lim_(n→+∞) ∫_0 ^n (1−(t/n))^n arctan(nt)dt

$$\mathrm{calculate}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\mathrm{n}} \left(\mathrm{1}−\frac{\mathrm{t}}{\mathrm{n}}\right)^{\mathrm{n}} \mathrm{arctan}\left(\mathrm{nt}\right)\mathrm{dt} \\ $$

Question Number 98944 Answers: 1 Comments: 0

let g(x) =((cosx +1)/(cos(2x)−3)) developp f at fourier serie

$$\mathrm{let}\:\mathrm{g}\left(\mathrm{x}\right)\:=\frac{\mathrm{cosx}\:+\mathrm{1}}{\mathrm{cos}\left(\mathrm{2x}\right)−\mathrm{3}}\:\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$

Question Number 98943 Answers: 2 Comments: 0

let f(x) =(1/((1+x^2 )^3 )) developp f at integr serie

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$

Question Number 98942 Answers: 3 Comments: 2

calculate ∫ ((x+1−(√(2x+3)))/(x−2 +(√(x+1)))) dx

$$\mathrm{calculate}\:\int\:\frac{\mathrm{x}+\mathrm{1}−\sqrt{\mathrm{2x}+\mathrm{3}}}{\mathrm{x}−\mathrm{2}\:+\sqrt{\mathrm{x}+\mathrm{1}}}\:\mathrm{dx} \\ $$

Question Number 98940 Answers: 2 Comments: 1

if (1/((1+x)^n )) =Σ_(m=0) ^∞ a_m x^m determinate a_m

$$\mathrm{if}\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{n}} }\:=\sum_{\mathrm{m}=\mathrm{0}} ^{\infty} \:\mathrm{a}_{\mathrm{m}} \mathrm{x}^{\mathrm{m}} \:\:\:\:\mathrm{determinate}\:\mathrm{a}_{\mathrm{m}} \\ $$

Question Number 98932 Answers: 0 Comments: 0

Question Number 98929 Answers: 0 Comments: 6

Find[]the[]integral[]of[] ∫(dt/(√((1+t^(10) ))))

$${Find}\left[\right]{the}\left[\right]{integral}\left[\right]{of}\left[\right] \\ $$$$ \\ $$$$\int\frac{{dt}}{\sqrt{\left(\mathrm{1}+{t}^{\mathrm{10}} \right)}} \\ $$

Question Number 98925 Answers: 1 Comments: 1

If the curve shown below has the equation, y=(x−p)(x^3 −bx−c) then find q/p in terms of b and c.

$${If}\:{the}\:{curve}\:{shown}\:{below}\:{has}\:{the}\: \\ $$$${equation},\:\:{y}=\left({x}−{p}\right)\left({x}^{\mathrm{3}} −{bx}−{c}\right) \\ $$$${then}\:{find}\:\:{q}/{p}\:\:{in}\:{terms}\:{of}\:{b}\:{and}\:{c}. \\ $$

Question Number 98924 Answers: 0 Comments: 0

Prove that if a+ bi is a root to pz^2 + qz + r = 0 , where a,b,p,q,r ∈R then a−bi is also a root to that equation.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+\:{bi}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{to} \\ $$$$\:{pz}^{\mathrm{2}} \:+\:{qz}\:+\:{r}\:=\:\mathrm{0}\:,\:\mathrm{where}\:{a},{b},{p},{q},{r}\:\in\mathbb{R} \\ $$$$\mathrm{then}\:{a}−{bi}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}\:\mathrm{to}\:\mathrm{that}\:\mathrm{equation}. \\ $$

Question Number 98938 Answers: 2 Comments: 0

Question Number 98919 Answers: 0 Comments: 1

Question Number 98914 Answers: 3 Comments: 2

2x+3y=5 (x^2 +y^2 )_(min) =?

$$\mathrm{2}{x}+\mathrm{3}{y}=\mathrm{5} \\ $$$$\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \underset{{min}} {\right)}=? \\ $$

Question Number 98913 Answers: 0 Comments: 3

solve f ′(x)=f(f(x))

$${solve} \\ $$$${f}\:'\left({x}\right)={f}\left({f}\left({x}\right)\right) \\ $$

Question Number 98907 Answers: 1 Comments: 0

Question Number 98895 Answers: 2 Comments: 0

Is (dy/dx) if y^3 +x^3 −2x=1 ?

$${Is}\:\frac{{dy}}{{dx}}\:{if}\:{y}^{\mathrm{3}} +{x}^{\mathrm{3}} −\mathrm{2}{x}=\mathrm{1}\:?\: \\ $$

Question Number 99173 Answers: 1 Comments: 0

Question Number 98887 Answers: 0 Comments: 0

Question Number 98885 Answers: 0 Comments: 2

find the range f(x)=log_4 log_2 log_(1/2) (x)

$${find}\:{the}\:{range} \\ $$$$ \\ $$$${f}\left({x}\right)={log}_{\mathrm{4}} {log}_{\mathrm{2}} {log}_{\frac{\mathrm{1}}{\mathrm{2}}} \left({x}\right) \\ $$

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