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Question Number 156517 Answers: 1 Comments: 0

∫ (dx/(3tan x+cos x)) =?

$$\int\:\frac{\mathrm{dx}}{\mathrm{3tan}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}\:=? \\ $$

Question Number 156509 Answers: 1 Comments: 5

Question Number 156508 Answers: 1 Comments: 0

Question Number 156507 Answers: 0 Comments: 0

Question Number 156502 Answers: 1 Comments: 0

Solve for integers: (x^2 + y^2 )(x^4 + y^4 ) = (x + y)^6

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{integers}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \right)\left(\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \right)\:=\:\left(\mathrm{x}\:+\:\mathrm{y}\right)^{\mathrm{6}} \\ $$$$ \\ $$

Question Number 156495 Answers: 1 Comments: 0

1. Write the first three terms of each of the following sequences: (a) a_n = (3^n /(2^n + 1)) (b) a_n = (−1)^(n−1) n^3

$$\mathrm{1}.\:\mathrm{Write}\:\mathrm{the}\:\mathrm{first}\:\mathrm{three}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sequences}:\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{a}_{{n}} \:=\:\frac{\mathrm{3}^{{n}} }{\mathrm{2}^{{n}} \:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{a}_{{n}} \:=\:\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} {n}^{\mathrm{3}} \\ $$

Question Number 156485 Answers: 0 Comments: 0

2021! Σ_(n=0) ^∞ Σ_(k=0) ^∞ (1/(Π_(m=1) ^(2022) (n + k + m))) = (1/A) Find the value of A

$$\mathrm{2021}!\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\underset{\boldsymbol{\mathrm{m}}=\mathrm{1}} {\overset{\mathrm{2022}} {\prod}}\left(\mathrm{n}\:+\:\mathrm{k}\:+\:\mathrm{m}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{A}} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\boldsymbol{\mathrm{A}} \\ $$

Question Number 156475 Answers: 1 Comments: 4

soit:∫_o ^x f(t)=x+∫_0 ^x t^2 f(t)dt determiner f(1)

$${soit}:\int_{{o}} ^{{x}} {f}\left({t}\right)={x}+\int_{\mathrm{0}} ^{{x}} {t}^{\mathrm{2}} {f}\left({t}\right){dt} \\ $$$${determiner}\:{f}\left(\mathrm{1}\right) \\ $$

Question Number 156470 Answers: 1 Comments: 0

Q=∫_0 ^∞ (x^c /c^x )dx

$${Q}=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{c}} }{{c}^{{x}} }{dx} \\ $$

Question Number 156473 Answers: 1 Comments: 0

Question Number 156467 Answers: 0 Comments: 0

if x;y;z≥0 and 4xyz+4xy+2yz+3zx=6 prove that: 2x+3y+4z ≥ 4(xy+yz+zx)

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\geqslant\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{4xyz}+\mathrm{4xy}+\mathrm{2yz}+\mathrm{3zx}=\mathrm{6} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{2x}+\mathrm{3y}+\mathrm{4z}\:\geqslant\:\mathrm{4}\left(\mathrm{xy}+\mathrm{yz}+\mathrm{zx}\right) \\ $$

Question Number 156463 Answers: 1 Comments: 0

Find: f : N^∗ → N^∗ ; 2(f(x) + f(y)) = x ∀x∈N^∗

$$\mathrm{Find}: \\ $$$$\mathrm{f}\::\:\mathbb{N}^{\ast} \:\rightarrow\:\mathbb{N}^{\ast} \:\:;\:\:\:\mathrm{2}\left(\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{y}\right)\right)\:=\:\mathrm{x} \\ $$$$\forall\mathrm{x}\in\mathbb{N}^{\ast} \\ $$

Question Number 156462 Answers: 1 Comments: 0

If abc^(−) = cba^(−) + def^(−) and a∈{c+2;...;9} Then find def^(−) + fed^(−)

$$\mathrm{If}\:\:\:\overline {\mathrm{abc}}\:=\:\overline {\mathrm{cba}}\:+\:\overline {\mathrm{def}}\:\:\mathrm{and}\:\:\mathrm{a}\in\left\{\mathrm{c}+\mathrm{2};...;\mathrm{9}\right\} \\ $$$$\mathrm{Then}\:\mathrm{find}\:\:\:\overline {\mathrm{def}}\:+\:\overline {\mathrm{fed}}\: \\ $$$$ \\ $$

Question Number 156482 Answers: 1 Comments: 0

Solve for real numbers: 3 + sin(2x) = 4sin(x + (π/4))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{3}\:+\:\mathrm{sin}\left(\mathrm{2x}\right)\:=\:\mathrm{4sin}\left(\mathrm{x}\:+\:\frac{\pi}{\mathrm{4}}\right) \\ $$

Question Number 156573 Answers: 0 Comments: 0

if a;b;c;d>1 then: Σ_(cyc) log_a (((b^3 + c^3 + d^3 )/(b^2 + c^2 + d^2 ))) ≥ 4

$$\mathrm{if}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d}>\mathrm{1}\:\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\mathrm{log}_{\boldsymbol{\mathrm{a}}} \:\left(\frac{\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \:+\:\mathrm{d}^{\mathrm{3}} }{\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:+\:\mathrm{d}^{\mathrm{2}} }\right)\:\geqslant\:\mathrm{4} \\ $$

Question Number 156571 Answers: 1 Comments: 0

Question Number 156458 Answers: 0 Comments: 0

Question Number 156455 Answers: 1 Comments: 4

Question Number 156453 Answers: 0 Comments: 0

Solve for real numbers: 9 + log_2 (x/(x^4 + 48)) = 2 + (2/( (√(x - 1))))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\mathrm{9}\:+\:\mathrm{log}_{\mathrm{2}} \:\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{48}}\:=\:\mathrm{2}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{x}\:-\:\mathrm{1}}} \\ $$

Question Number 156452 Answers: 0 Comments: 0

Find x;y;z≥0 such that: { ((x-y-z = sinx-siny-sinz)),((x^2 -y^2 -z^2 = sin^2 x-sin^2 y-sin^2 z)),((x^3 -y^3 -z^3 = sin^3 x-sin^3 y-sin^3 z)) :}

Question Number 156450 Answers: 0 Comments: 2

Question Number 156448 Answers: 1 Comments: 0

If 0<a≤b<π then prove that: ((sin(√(ab)))/(sin(((a+b)/2)))) ≥ ((32a^2 b^2 (√(ab)))/((a+b)^5 ))

$$\mathrm{If}\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b}<\pi\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{sin}\sqrt{\mathrm{ab}}}{\mathrm{sin}\left(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}\right)}\:\geqslant\:\frac{\mathrm{32a}^{\mathrm{2}} \mathrm{b}^{\mathrm{2}} \sqrt{\mathrm{ab}}}{\left(\mathrm{a}+\mathrm{b}\right)^{\mathrm{5}} } \\ $$

Question Number 156447 Answers: 1 Comments: 0

Find all functions f : Z → R such that f(n+m)=nf(n)+mf(m)+nm-n-m ∀n;m∈Z

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{functions}\:\:\mathrm{f}\::\:\mathbb{Z}\:\rightarrow\:\mathbb{R}\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{n}+\mathrm{m}\right)=\mathrm{nf}\left(\mathrm{n}\right)+\mathrm{mf}\left(\mathrm{m}\right)+\mathrm{nm}-\mathrm{n}-\mathrm{m} \\ $$$$\forall\mathrm{n};\mathrm{m}\in\mathbb{Z} \\ $$

Question Number 156434 Answers: 0 Comments: 0

Question Number 156432 Answers: 0 Comments: 0

in a second order differenti equation, a 4 henry inductor,an 8 ohm resistor and o.2 farad capacito are connected in series with the temperature of the battery with ggl. E=80 sin 3t. solid=0 the charge in the capacitor and the current in the circuit is zero. a.charge b.current at t>0

$$\mathrm{in}\:\mathrm{a}\:\mathrm{second}\:\mathrm{order}\:\mathrm{differenti}\:\mathrm{equation},\:\mathrm{a}\:\mathrm{4}\:\mathrm{henry} \\ $$$$\mathrm{inductor},\mathrm{an}\:\mathrm{8}\:\mathrm{ohm}\:\mathrm{resistor}\:\mathrm{and}\:\mathrm{o}.\mathrm{2}\:\mathrm{farad}\:\mathrm{capacito} \\ $$$$\mathrm{are}\:\mathrm{connected}\:\mathrm{in}\:\mathrm{series}\:\mathrm{with}\:\mathrm{the}\:\mathrm{temperature}\:\mathrm{of}\: \\ $$$$\mathrm{the}\:\mathrm{battery}\:\mathrm{with}\:\mathrm{ggl}.\:\mathrm{E}=\mathrm{80}\:\mathrm{sin}\:\mathrm{3t}.\:\mathrm{solid}=\mathrm{0}\:\mathrm{the} \\ $$$$\mathrm{charge}\:\mathrm{in}\:\mathrm{the}\:\mathrm{capacitor}\:\mathrm{and}\:\mathrm{the}\:\mathrm{current}\:\mathrm{in}\:\mathrm{the}\: \\ $$$$\mathrm{circuit}\:\mathrm{is}\:\mathrm{zero}. \\ $$$$\mathrm{a}.\mathrm{charge} \\ $$$$\mathrm{b}.\mathrm{current}\:\mathrm{at}\:\mathrm{t}>\mathrm{0} \\ $$

Question Number 156426 Answers: 1 Comments: 0

Given that tan α and tan β are the roots of the equation x^2 +3ax+4a+1=0, where a>1 and α,β∈(−(π/2),(π/2)). Evaluate tan(((α+β)/2)).

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{tan}\:\alpha\:\mathrm{and}\:\mathrm{tan}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} +\mathrm{3}{ax}+\mathrm{4}{a}+\mathrm{1}=\mathrm{0},\: \\ $$$$\mathrm{where}\:{a}>\mathrm{1}\:\mathrm{and}\:\alpha,\beta\in\left(−\frac{\pi}{\mathrm{2}},\frac{\pi}{\mathrm{2}}\right).\: \\ $$$$\mathrm{Evaluate}\:\mathrm{tan}\left(\frac{\alpha+\beta}{\mathrm{2}}\right). \\ $$

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