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Question Number 60287 Answers: 2 Comments: 2

f(x) = x^3 + 3x − 7 f^(−1) (x) = ?

$${f}\left({x}\right)\:\:=\:\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}\:−\:\mathrm{7} \\ $$$${f}\:^{−\mathrm{1}} \left({x}\right)\:\:=\:\:? \\ $$

Question Number 60283 Answers: 1 Comments: 1

Question Number 60269 Answers: 3 Comments: 3

if tan A − cot A = 0 prove that sin A + cos A=?

$${if} \\ $$$${tan}\:{A}\:−\:{cot}\:{A}\:=\:\mathrm{0} \\ $$$${prove}\:{that} \\ $$$${sin}\:{A}\:+\:{cos}\:{A}=? \\ $$

Question Number 60264 Answers: 0 Comments: 0

let f(t) =∫_0 ^∞ (e^(−3 [x^2 ]) /(x^2 +t^2 ))dx with t>0 1. determine a explicit form of f(t) 2. find also g(t) =∫_0 ^∞ (e^(−3[x^2 ]) /((x^2 +t^2 )^2 ))dx 3. find the values of integrals ∫_0 ^∞ (e^(−3[x^2 ]) /(x^2 +3))dx and ∫_0 ^∞ (e^(−3[x^2 ]) /((x^2 +4)^2 )) dx .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−\mathrm{3}\:\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dx}\:\:{with}\:{t}>\mathrm{0} \\ $$$$\mathrm{1}.\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\mathrm{2}.\:{find}\:{also}\:{g}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\mathrm{3}.\:{find}\:{the}\:{values}\:{of}\:{integrals}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−\mathrm{3}\left[{x}^{\mathrm{2}} \right]} }{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 60263 Answers: 0 Comments: 1

let U_n =∫_0 ^∞ (e^(−n[x^2 ]) /(x^2 +3)) dx 1) calculate U_n interms of n 2) find lim_(n→+∞) n U_n 3)determine nature of the serie Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{e}^{−{n}\left[{x}^{\mathrm{2}} \right]} }{{x}^{\mathrm{2}} +\mathrm{3}}\:{dx}\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right){determine}\:{nature}\:{of}\:{the}\:{serie}\:\:\Sigma\:{U}_{{n}} \\ $$

Question Number 60257 Answers: 0 Comments: 4

The maximum value of Z=4x+2y subject to constraints 2x+3y≤18 , x+y≥10 and x,y≥0 is ?

$${The}\:{maximum}\:{value}\:{of}\:{Z}=\mathrm{4}{x}+\mathrm{2}{y} \\ $$$${subject}\:{to}\:{constraints}\:\mathrm{2}{x}+\mathrm{3}{y}\leqslant\mathrm{18}\:, \\ $$$${x}+{y}\geqslant\mathrm{10}\:{and}\:{x},{y}\geqslant\mathrm{0}\:{is}\:? \\ $$

Question Number 60256 Answers: 1 Comments: 0

Question Number 60255 Answers: 0 Comments: 0

b=(((kT)/P))^(1/3) . distance molekular prove.

$$\boldsymbol{\mathrm{b}}=\sqrt[{\mathrm{3}}]{\frac{\boldsymbol{\mathrm{kT}}}{\boldsymbol{\mathrm{P}}}}.\:\boldsymbol{\mathrm{distance}}\:\:\boldsymbol{\mathrm{molekular}} \\ $$$$\boldsymbol{\mathrm{prove}}. \\ $$

Question Number 60275 Answers: 0 Comments: 0

Question Number 60274 Answers: 0 Comments: 1

Question Number 60247 Answers: 0 Comments: 0

A uniform pole PQ, 30 m long and of mass 4 kg is carried by a boy at P and a man 8 m away from Q. Find the distance from P where a mass of 20 kg should be attached so that the man′s support is twice that of the boy, if the system is in equilibrium [Take g=10ms^(−2) ]

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{pole}\:\mathrm{PQ},\:\mathrm{30}\:\mathrm{m}\:\mathrm{long}\:\mathrm{and}\:\mathrm{of}\:\mathrm{mass} \\ $$$$\mathrm{4}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{carried}\:\mathrm{by}\:\mathrm{a}\:\mathrm{boy}\:\mathrm{at}\:\mathrm{P}\:\mathrm{and}\:\mathrm{a}\:\mathrm{man}\: \\ $$$$\mathrm{8}\:\mathrm{m}\:\mathrm{away}\:\mathrm{from}\:\mathrm{Q}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{from} \\ $$$$\mathrm{P}\:\mathrm{where}\:\mathrm{a}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{20}\:\mathrm{kg}\:\mathrm{should}\:\mathrm{be}\:\mathrm{attached} \\ $$$$\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{man}'\mathrm{s}\:\mathrm{support}\:\mathrm{is}\:\mathrm{twice}\:\mathrm{that} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{boy},\:\mathrm{if}\:\mathrm{the}\:\mathrm{system}\:\mathrm{is}\:\mathrm{in}\:\mathrm{equilibrium} \\ $$$$\left[\mathrm{Take}\:\mathrm{g}=\mathrm{10ms}^{−\mathrm{2}} \right] \\ $$

Question Number 60237 Answers: 0 Comments: 1

Question Number 60234 Answers: 1 Comments: 0

Question Number 60227 Answers: 0 Comments: 0

Question Number 60219 Answers: 1 Comments: 1

C=((2𝛑𝛜𝛜_0 L)/(ln((R_2 /R_1 )))). prove.

$$\boldsymbol{\mathrm{C}}=\frac{\mathrm{2}\boldsymbol{\pi\varepsilon\varepsilon}_{\mathrm{0}} \boldsymbol{\mathrm{L}}}{\boldsymbol{\mathrm{ln}}\left(\frac{\boldsymbol{\mathrm{R}}_{\mathrm{2}} }{\boldsymbol{\mathrm{R}}_{\mathrm{1}} }\right)}. \\ $$$$\boldsymbol{\mathrm{prove}}. \\ $$

Question Number 60241 Answers: 0 Comments: 1

Question Number 60240 Answers: 0 Comments: 1

∫_0 ^2 ((ln(x))/(√(4−x^2 )))dx

$$\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\frac{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{x}}\right)}{\sqrt{\mathrm{4}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 60238 Answers: 0 Comments: 0

Question Number 60212 Answers: 1 Comments: 1

Question Number 60203 Answers: 2 Comments: 1

demonstrate ∣sin(y)−sin(x)∣≤∣y−x∣

$${demonstrate}\: \\ $$$$\mid{sin}\left({y}\right)−{sin}\left({x}\right)\mid\leqslant\mid{y}−{x}\mid \\ $$

Question Number 60202 Answers: 0 Comments: 0

construct the point M^′ =(1/2)((((z+∣z∣))/2))

$${construct}\:{the}\:{point}\:{M}^{'} =\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\left({z}+\mid{z}\mid\right)}{\mathrm{2}}\right) \\ $$

Question Number 60197 Answers: 1 Comments: 1

valculste lim_(n→+∞) (ln((Π_(k=1) ^n (1+(k^3 /n^3 )))^(1/n) )

$${valculste}\:{lim}_{{n}\rightarrow+\infty} \left({ln}\left(\left(\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{{k}^{\mathrm{3}} }{{n}^{\mathrm{3}} }\right)\right)^{\frac{\mathrm{1}}{{n}}} \right)\right. \\ $$

Question Number 60198 Answers: 1 Comments: 0

find lim_(n→+∞) ln(Π_(k=1) ^n (1+(k^4 /n^4 ))^(1/n) )

$${find}\:{lim}_{{n}\rightarrow+\infty} \:{ln}\left(\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{{k}^{\mathrm{4}} }{{n}^{\mathrm{4}} }\right)^{\frac{\mathrm{1}}{{n}}} \right) \\ $$

Question Number 60189 Answers: 0 Comments: 0

Question Number 60175 Answers: 0 Comments: 2

solving u^v =w with u, v, w ∈C finding all possible solutions I tested this with several values and found no mistake. please review and comment. I hope this will help at least some of you.

$$\mathrm{solving}\:{u}^{{v}} ={w}\:\mathrm{with}\:{u},\:{v},\:{w}\:\in\mathbb{C} \\ $$$$\mathrm{finding}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solutions} \\ $$$$\mathrm{I}\:\mathrm{tested}\:\mathrm{this}\:\mathrm{with}\:\mathrm{several}\:\mathrm{values}\:\mathrm{and}\:\mathrm{found} \\ $$$$\mathrm{no}\:\mathrm{mistake}.\:\mathrm{please}\:\mathrm{review}\:\mathrm{and}\:\mathrm{comment}. \\ $$$$\mathrm{I}\:\mathrm{hope}\:\mathrm{this}\:\mathrm{will}\:\mathrm{help}\:\mathrm{at}\:\mathrm{least}\:\mathrm{some}\:\mathrm{of}\:\mathrm{you}. \\ $$

Question Number 60170 Answers: 0 Comments: 0

∫xsec^3 xdx

$$\int{x}\mathrm{sec}\:^{\mathrm{3}} {xdx} \\ $$

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