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Question Number 105285 Answers: 1 Comments: 2

simplify ((sin θ)/(1+cot θ)) + ((cos θ)/(1+ tan θ)) ?

$${simplify}\:\frac{\mathrm{sin}\:\theta}{\mathrm{1}+\mathrm{cot}\:\theta}\:+\:\frac{\mathrm{cos}\:\theta}{\mathrm{1}+\:\mathrm{tan}\:\theta}\:? \\ $$

Question Number 102146 Answers: 0 Comments: 14

I decided it′s time to leave for a while. I need some peace and calm, I′m losing my zenter... I′ll put more time and energy into my music. I guess I′ll be back; until then, kindest regards & thanks for the teachings!

$$\mathrm{I}\:\mathrm{decided}\:\mathrm{it}'\mathrm{s}\:\mathrm{time}\:\mathrm{to}\:\mathrm{leave}\:\mathrm{for}\:\mathrm{a}\:\mathrm{while}.\:\mathrm{I}\:\mathrm{need} \\ $$$$\mathrm{some}\:\mathrm{peace}\:\mathrm{and}\:\mathrm{calm},\:\mathrm{I}'\mathrm{m}\:\mathrm{losing}\:\mathrm{my}\:{zen}\mathrm{ter}... \\ $$$$\mathrm{I}'\mathrm{ll}\:\mathrm{put}\:\mathrm{more}\:\mathrm{time}\:\mathrm{and}\:\mathrm{energy}\:\mathrm{into}\:\mathrm{my}\:\mathrm{music}. \\ $$$$\mathrm{I}\:\mathrm{guess}\:\mathrm{I}'\mathrm{ll}\:\mathrm{be}\:\mathrm{back};\:\mathrm{until}\:\mathrm{then}, \\ $$$$\mathrm{kindest}\:\mathrm{regards}\:\&\:\mathrm{thanks}\:\mathrm{for}\:\mathrm{the}\:\mathrm{teachings}! \\ $$

Question Number 105283 Answers: 3 Comments: 0

(1) (dy/dx) = ((2xy)/(4x^2 −y^3 )) (2) (dy/dx) = ((sin x+cos x)/(y(2ln y + 1)))

$$\left(\mathrm{1}\right)\:\frac{{dy}}{{dx}}\:=\:\frac{\mathrm{2}{xy}}{\mathrm{4}{x}^{\mathrm{2}} −{y}^{\mathrm{3}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{{dy}}{{dx}}\:=\:\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{{y}\left(\mathrm{2ln}\:{y}\:+\:\mathrm{1}\right)} \\ $$

Question Number 102142 Answers: 0 Comments: 0

A four-sidex dice with numbered 1, 2, 3, and 4 is thrown and the number at the base is read. The dice is biased such that the probabilies P_1 , P_2 , P_3 , and P_4 to obtain 1, 2, 3, and 4 respectively are in an arithmetic progression. 1\ Given P_4 =0.4, calculate P_1 , P_2 , and P_3 . 2\ The dice is thrown n-times (n≥1). The throws are assumed to be independent, 2 by 2, and identical. Given U_n -the probability of obtaining for the first time the fourth-n^(th) throw; a\Express U_n in terms of n= b\Given S_n =Σ_(i=1) ^n U_i i. Express Sn in terms of n, and find its limit. ii. Determine the smallest natural number such that S_n >0.999

$$\mathrm{A}\:\mathrm{four}-\mathrm{sidex}\:\mathrm{dice}\:\mathrm{with}\:\mathrm{numbered}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{and}\:\mathrm{4}\:\mathrm{is}\:\mathrm{thrown} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{number}\:\mathrm{at}\:\mathrm{the}\:\mathrm{base}\:\mathrm{is}\:\mathrm{read}. \\ $$$$\mathrm{The}\:\mathrm{dice}\:\mathrm{is}\:\mathrm{biased}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{probabilies}\:\mathrm{P}_{\mathrm{1}} ,\:\mathrm{P}_{\mathrm{2}} ,\:\mathrm{P}_{\mathrm{3}} , \\ $$$$\mathrm{and}\:\mathrm{P}_{\mathrm{4}} \:\mathrm{to}\:\mathrm{obtain}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{and}\:\mathrm{4}\:\mathrm{respectively}\:\mathrm{are}\:\mathrm{in}\:\mathrm{an} \\ $$$$\mathrm{arithmetic}\:\mathrm{progression}. \\ $$$$\mathrm{1}\backslash\:\mathrm{Given}\:\mathrm{P}_{\mathrm{4}} =\mathrm{0}.\mathrm{4},\:\mathrm{calculate}\:\mathrm{P}_{\mathrm{1}} ,\:\mathrm{P}_{\mathrm{2}} ,\:\mathrm{and}\:\mathrm{P}_{\mathrm{3}} . \\ $$$$\mathrm{2}\backslash\:\mathrm{The}\:\mathrm{dice}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{n}-\mathrm{times}\:\left(\mathrm{n}\geqslant\mathrm{1}\right).\:\mathrm{The}\:\mathrm{throws}\:\mathrm{are}\:\mathrm{assumed} \\ $$$$\mathrm{to}\:\mathrm{be}\:\mathrm{independent},\:\mathrm{2}\:\mathrm{by}\:\mathrm{2},\:\mathrm{and}\:\mathrm{identical}.\:\mathrm{Given}\:\mathrm{U}_{\mathrm{n}} -\mathrm{the}\:\mathrm{probability} \\ $$$$\mathrm{of}\:\mathrm{obtaining}\:\mathrm{for}\:\mathrm{the}\:\mathrm{first}\:\mathrm{time}\:\mathrm{the}\:\mathrm{fourth}-\mathrm{n}^{\mathrm{th}} \mathrm{throw}; \\ $$$$\mathrm{a}\backslash\mathrm{Express}\:\mathrm{U}_{\mathrm{n}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{n}= \\ $$$$\mathrm{b}\backslash\mathrm{Given}\:\mathrm{S}_{\mathrm{n}} =\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{U}_{\mathrm{i}} \\ $$$$\:\:\mathrm{i}.\:\mathrm{Express}\:\mathrm{Sn}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{n},\:\mathrm{and}\:\mathrm{find}\:\mathrm{its}\:\mathrm{limit}. \\ $$$$\:\mathrm{ii}.\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{natural}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\mathrm{S}_{\mathrm{n}} >\mathrm{0}.\mathrm{999} \\ $$

Question Number 102139 Answers: 1 Comments: 1

We have 4 boys and 2 girls. we choose at random and simultaneous 2 boys and 1 girl to for a group. 1) How many possibilities do we have?

$${We}\:{have}\:\mathrm{4}\:{boys}\:{and}\:\mathrm{2}\:{girls}.\:{we} \\ $$$${choose}\:{at}\:{random}\:{and}\:{simultaneous} \\ $$$$\mathrm{2}\:{boys}\:{and}\:\mathrm{1}\:{girl}\:{to}\:{for}\:{a}\:{group}. \\ $$$$\left.\mathrm{1}\right)\:{How}\:{many}\:{possibilities}\:{do}\:{we}\:{have}? \\ $$

Question Number 102138 Answers: 0 Comments: 0

∫_0 ^1 ∫_(√y) ^0 (√(x^3 +1+ax^2 +bx+c)) dx dy

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\sqrt{{y}}} ^{\mathrm{0}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}+{ax}^{\mathrm{2}} +{bx}+{c}}\:\:{dx}\:{dy} \\ $$

Question Number 102196 Answers: 0 Comments: 0

ABCD is a square with center O. AB=5 cm. EFGH is a square such as AE=BF=CG=DH. 1) Show with Chasles rules that EF^(→) .FG^(→) =0^→ 2) Determinate the area of EBFO according to the area of ABCD.

$${ABCD}\:{is}\:{a}\:{square}\:{with}\:{center}\:{O}. \\ $$$${AB}=\mathrm{5}\:{cm}. \\ $$$${EFGH}\:{is}\:{a}\:{square}\:{such}\:{as}\: \\ $$$${AE}={BF}={CG}={DH}. \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{with}\:{Chasles}\:{rules}\:{that} \\ $$$$\overset{\rightarrow} {{EF}}.\overset{\rightarrow} {{FG}}=\overset{\rightarrow} {\mathrm{0}} \\ $$$$\left.\mathrm{2}\right)\:{Determinate}\:{the}\:{area}\:{of}\:{EBFO} \\ $$$${according}\:{to}\:{the}\:{area}\:{of}\:{ABCD}. \\ $$

Question Number 102128 Answers: 4 Comments: 0

∫((x+1)/((√x) +1))dx=?

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

Question Number 102121 Answers: 3 Comments: 1

∫_(−∞) ^∞ ((sin(x+(π/2)))/(1+x^2 )) dx By real analysis

$$\int_{−\infty} ^{\infty} \frac{{sin}\left({x}+\frac{\pi}{\mathrm{2}}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{By}\:{real}\:{analysis} \\ $$

Question Number 102115 Answers: 3 Comments: 3

Γ(s)ζ(s)=∫_0 ^∞ (x^(s−1) /(e^x +1))dx (Prove that) And prove 1+2+3+4+5+6+7+....∞=−(1/(12))

$$\Gamma\left({s}\right)\zeta\left({s}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{s}−\mathrm{1}} }{{e}^{{x}} +\mathrm{1}}{dx}\:\:\left({Prove}\:{that}\right) \\ $$$${And}\:{prove}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+....\infty=−\frac{\mathrm{1}}{\mathrm{12}} \\ $$$$ \\ $$

Question Number 102127 Answers: 1 Comments: 0

∫_0 ^2 (1−x^2 )^3 dx=? and write the furmollah

$$\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{3}} {dx}=? \\ $$$$\:{and}\:{write}\:{the}\:{furmollah} \\ $$

Question Number 102109 Answers: 0 Comments: 0

Question Number 102101 Answers: 1 Comments: 1

We have 2 red and 1 black tokens. We want to dispose them on 9 places of this checker−board. 1)What the number of possibilities to dispose these 3 tokens on these 9 places. 2)What it is the number of possibilities to dispose these 3 tokens like that: a) one red token on one black place. b)two red tokens on black places. c)zero red token one black places.

$${We}\:{have}\:\mathrm{2}\:{red}\:{and}\:\mathrm{1}\:{black}\:{tokens}. \\ $$$${We}\:{want}\:{to}\:{dispose}\:{them}\:{on}\:\mathrm{9}\:{places} \\ $$$${of}\:{this}\:{checker}−{board}. \\ $$$$\left.\mathrm{1}\right){What}\:{the}\:{number}\:{of}\:{possibilities}\: \\ $$$${to}\:{dispose}\:{these}\:\mathrm{3}\:{tokens}\:{on}\:{these} \\ $$$$\mathrm{9}\:{places}. \\ $$$$\left.\mathrm{2}\right){What}\:{it}\:{is}\:{the}\:{number}\:{of}\:{possibilities} \\ $$$${to}\:{dispose}\:{these}\:\mathrm{3}\:{tokens}\:\:{like}\:{that}: \\ $$$$\left.{a}\right)\:{one}\:{red}\:{token}\:{on}\:{one}\:{black}\:{place}. \\ $$$$\left.{b}\right){two}\:{red}\:{tokens}\:{on}\:{black}\:{places}. \\ $$$$\left.{c}\right){zero}\:{red}\:{token}\:{one}\:{black}\:{places}. \\ $$$$ \\ $$

Question Number 102099 Answers: 1 Comments: 0

∫sinx ∙ cosx ∙cos2x ∙ cos4x dx=?

$$\int{sinx}\:\centerdot\:{cosx}\:\centerdot{cos}\mathrm{2}{x}\:\centerdot\:{cos}\mathrm{4}{x}\:{dx}=? \\ $$

Question Number 102097 Answers: 0 Comments: 2

∫((cos^2 x −cos^2 x)/(cosx−cosx))dx=?

$$\int\frac{{cos}^{\mathrm{2}} {x}\:−{cos}^{\mathrm{2}} {x}}{{cosx}−{cosx}}{dx}=? \\ $$

Question Number 102096 Answers: 0 Comments: 8

Question Number 102094 Answers: 0 Comments: 5

good evenig for all this is an answerd question i will repost it if: f(x)=(√(x−2)) is there a cirtical point when x=2 ?

$${good}\:{evenig}\:{for}\:{all} \\ $$$$ \\ $$$${this}\:{is}\:{an}\:{answerd}\:{question}\:{i}\:{will}\:{repost}\:{it} \\ $$$${if}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{f}\left({x}\right)=\sqrt{{x}−\mathrm{2}} \\ $$$${is}\:{there}\:{a}\:{cirtical}\:{point}\:{when}\:{x}=\mathrm{2}\:? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 102089 Answers: 2 Comments: 0

The Gamma function Γ(α) is defined as follows; Γ(α)=∫_0 ^∞ y^(α−1) e^(−y) dy , α>0 a\ Show that Γ(α+1)=αΓ(α). b\Conclude that Γ(n)=(n−1)! , n=1, 2, 3, ... c\Determine Γ(55).

$$ \\ $$$$\mathrm{The}\:\mathrm{Gamma}\:\mathrm{function}\:\Gamma\left(\alpha\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{follows}; \\ $$$$\Gamma\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \mathrm{y}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{y}} \mathrm{dy}\:,\:\alpha>\mathrm{0} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\Gamma\left(\alpha+\mathrm{1}\right)=\alpha\Gamma\left(\alpha\right). \\ $$$$\mathrm{b}\backslash\mathrm{Conclude}\:\mathrm{that}\:\Gamma\left(\mathrm{n}\right)=\left(\mathrm{n}−\mathrm{1}\right)!\:,\:\mathrm{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:... \\ $$$$\mathrm{c}\backslash\mathrm{Determine}\:\Gamma\left(\mathrm{55}\right). \\ $$

Question Number 102086 Answers: 1 Comments: 2

Question Number 102085 Answers: 0 Comments: 0

Question Number 102080 Answers: 0 Comments: 0

A random variable, X, has a Gamma distribution with parameters α and β, (α, β>0). The p.d.f has the form f(x)=(1/(Γ(α)β^α ))x^(n−1) e^(−x/β) , for x>0 , Γ(α)=(1/β^α )∫_0 ^∞ x^(α−1) e^(−x) dx a\ Show that the Gamma density is a proper p.d.f. b\Find the mean, variance, and moment-generating function of the Gamma distribution. c\Find the fourth moment using the definition of moments.

$$\mathrm{A}\:\mathrm{random}\:\mathrm{variable},\:\mathrm{X},\:\mathrm{has}\:\mathrm{a}\:\mathrm{Gamma}\:\mathrm{distribution}\:\mathrm{with} \\ $$$$\mathrm{parameters}\:\alpha\:\mathrm{and}\:\beta,\:\left(\alpha,\:\beta>\mathrm{0}\right).\:\mathrm{The}\:\mathrm{p}.\mathrm{d}.\mathrm{f}\:\mathrm{has}\:\mathrm{the}\:\mathrm{form} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\Gamma\left(\alpha\right)\beta^{\alpha} }\mathrm{x}^{\mathrm{n}−\mathrm{1}} \mathrm{e}^{−\mathrm{x}/\beta} ,\:\mathrm{for}\:\mathrm{x}>\mathrm{0}\:\:,\:\:\Gamma\left(\alpha\right)=\frac{\mathrm{1}}{\beta^{\alpha} }\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Gamma}\:\mathrm{density}\:\mathrm{is}\:\mathrm{a}\:\mathrm{proper}\:\mathrm{p}.\mathrm{d}.\mathrm{f}. \\ $$$$\mathrm{b}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{mean},\:\mathrm{variance},\:\mathrm{and}\:\mathrm{moment}-\mathrm{generating}\:\mathrm{function}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{Gamma}\:\mathrm{distribution}. \\ $$$$\mathrm{c}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{moment}\:\mathrm{using}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{moments}. \\ $$

Question Number 102078 Answers: 5 Comments: 0

Question Number 102076 Answers: 0 Comments: 4

A car is currently valued at $70350.00. If it loses 12% of its value at the beginning of each year, a) find its value after three and half years. b) find the depreciation after three years

$$\mathrm{A}\:\mathrm{car}\:\mathrm{is}\:\mathrm{currently}\:\mathrm{valued}\:\mathrm{at}\:\$\mathrm{70350}.\mathrm{00}. \\ $$$$\mathrm{If}\:\mathrm{it}\:\mathrm{loses}\:\mathrm{12\%}\:\mathrm{of}\:\mathrm{its}\:\mathrm{value}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{year}, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{its}\:\mathrm{value}\:\mathrm{after}\:\mathrm{three}\:\mathrm{and}\:\mathrm{half}\:\mathrm{years}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{depreciation}\:\mathrm{after}\:\mathrm{three}\:\mathrm{years} \\ $$

Question Number 102066 Answers: 3 Comments: 0

(1)∫ ((cos (ax) dx)/(√(sin ax−b))) (2) (D^3 +2D^2 +D)y = e^(2x) +x^2 −x (3)the area between the curves y = (2/x) and y = −x+3

$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{cos}\:\left(\mathrm{ax}\right)\:\mathrm{dx}}{\sqrt{\mathrm{sin}\:\mathrm{ax}−\mathrm{b}}} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{D}^{\mathrm{3}} +\mathrm{2D}^{\mathrm{2}} +\mathrm{D}\right)\mathrm{y}\:=\:\mathrm{e}^{\mathrm{2x}} +\mathrm{x}^{\mathrm{2}} −\mathrm{x} \\ $$$$\left(\mathrm{3}\right)\mathrm{the}\:\mathrm{area}\:\mathrm{between}\:\mathrm{the}\:\mathrm{curves} \\ $$$$\mathrm{y}\:=\:\frac{\mathrm{2}}{\mathrm{x}}\:\mathrm{and}\:\mathrm{y}\:=\:−\mathrm{x}+\mathrm{3}\: \\ $$

Question Number 102065 Answers: 1 Comments: 0

∫ ((x^2 dx)/((1−x)(√x))) ?

$$\int\:\frac{\mathrm{x}^{\mathrm{2}} \:\mathrm{dx}}{\left(\mathrm{1}−\mathrm{x}\right)\sqrt{\mathrm{x}}}\:? \\ $$

Question Number 102064 Answers: 1 Comments: 0

∫ ((sin 3x)/(cos 5x. cos 2x)) dx ?

$$\int\:\frac{\mathrm{sin}\:\mathrm{3x}}{\mathrm{cos}\:\mathrm{5x}.\:\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\:? \\ $$

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