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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 ?

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

Question Number 102060 Answers: 2 Comments: 1

Question Number 102043 Answers: 0 Comments: 1

∫e^(√(ax+b)) dx

$$\int{e}^{\sqrt{{ax}+{b}}} {dx} \\ $$

Question Number 102036 Answers: 3 Comments: 0

lim_(x→0) ((((x^2 −1))^(1/5) +((x+1))^(1/3) )/(((x−1))^(1/3) +(√(x+1))))=?

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{\sqrt[{\mathrm{5}}]{{x}^{\mathrm{2}} −\mathrm{1}}\:+\sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}}{\sqrt[{\mathrm{3}}]{{x}−\mathrm{1}}\:+\sqrt{{x}+\mathrm{1}}}=? \\ $$

Question Number 102034 Answers: 2 Comments: 0

lim_(x→2) ((((2x+4))^(1/3) −2)/(x^2 −x−2))=?

$${li}\underset{{x}\rightarrow\mathrm{2}} {{m}}\frac{\sqrt[{\mathrm{3}}]{\mathrm{2}{x}+\mathrm{4}}\:−\mathrm{2}}{{x}^{\mathrm{2}} −{x}−\mathrm{2}}=? \\ $$

Question Number 105269 Answers: 0 Comments: 0

a;b;c are real numbers 1<b<c^2 <a^(10) log_a b+2log_b c+5log_c a=12 prove that 2log_a c+5log_c b+10log_b a≥21

$${a};{b};{c}\:{are}\:{real}\:{numbers} \\ $$$$\mathrm{1}<{b}<{c}^{\mathrm{2}} <{a}^{\mathrm{10}} \\ $$$${log}_{{a}} {b}+\mathrm{2}{log}_{{b}} {c}+\mathrm{5}{log}_{{c}} {a}=\mathrm{12} \\ $$$${prove}\:{that} \\ $$$$\mathrm{2}{log}_{{a}} {c}+\mathrm{5}{log}_{{c}} {b}+\mathrm{10}{log}_{{b}} {a}\geqslant\mathrm{21} \\ $$

Question Number 105300 Answers: 0 Comments: 0

what is the absolute speed of A that moves via y = (1/(1+x)) (upper part) when it crosses y −axis meanwhile B moves at constant speed of 1 the x−axis ?

$${what}\:{is}\:{the}\:{absolute}\:{speed}\:{of}\:{A}\:{that} \\ $$$${moves}\:{via}\:{y}\:=\:\frac{\mathrm{1}}{\mathrm{1}+{x}}\:\left({upper}\:{part}\right)\:{when} \\ $$$${it}\:{crosses}\:{y}\:−{axis}\:{meanwhile}\:{B} \\ $$$${moves}\:{at}\:{constant}\:{speed}\:{of}\:\mathrm{1}\:{the} \\ $$$${x}−{axis}\:? \\ $$

Question Number 102020 Answers: 2 Comments: 0

lim_(x→0) (tan((π/4)−x))^(1/x)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left({tan}\left(\frac{\pi}{\mathrm{4}}−{x}\right)\right)^{\frac{\mathrm{1}}{{x}}} \\ $$

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