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Question Number 57948 Answers: 0 Comments: 0

let A(ξ) =∫_ξ ^ξ^2 ((arctan(1+ξt)−(π/4))/((√(2+ξt))−(√(2−ξt)))) dt find lim_(ξ →0) A(ξ) .

$${let}\:{A}\left(\xi\right)\:=\int_{\xi} ^{\xi^{\mathrm{2}} } \:\:\:\:\frac{{arctan}\left(\mathrm{1}+\xi{t}\right)−\frac{\pi}{\mathrm{4}}}{\sqrt{\mathrm{2}+\xi{t}}−\sqrt{\mathrm{2}−\xi{t}}}\:{dt} \\ $$$${find}\:{lim}_{\xi\:\rightarrow\mathrm{0}} \:\:{A}\left(\xi\right)\:. \\ $$$$ \\ $$

Question Number 57947 Answers: 0 Comments: 0

let P(x)=(1+ix)^n −1−ni with x real and n integr natural 1) find the roots of P(x) 2) factorize P(x) inside C[x] 3) factorize P(x) inside R[x] 4) decompose the fraction F(x) =((P^((1)) (x))/(P(x))) inside C(x) P^((1)) is the derivative of P .

$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{{n}} −\mathrm{1}−{ni}\:\:\:\:{with}\:{x}\:{real}\:{and}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:{factorize}\:{P}\left({x}\right)\:{inside}\:{R}\left[{x}\right] \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{{P}^{\left(\mathrm{1}\right)} \left({x}\right)}{{P}\left({x}\right)}\:{inside}\:{C}\left({x}\right) \\ $$$${P}^{\left(\mathrm{1}\right)} \:{is}\:{the}\:{derivative}\:{of}\:{P}\:. \\ $$

Question Number 57946 Answers: 0 Comments: 0

Question Number 57938 Answers: 0 Comments: 0

Question Number 57933 Answers: 0 Comments: 0

Question Number 57932 Answers: 1 Comments: 0

7+g=24

$$\mathrm{7}+{g}=\mathrm{24} \\ $$$$ \\ $$

Question Number 57931 Answers: 0 Comments: 0

What is 2^3 +6×4

$$\mathrm{What}\:\mathrm{is}\:\mathrm{2}^{\mathrm{3}} +\mathrm{6}×\mathrm{4} \\ $$

Question Number 57930 Answers: 1 Comments: 0

solve 2.3((2/(11))+3)

$$\mathrm{solve}\:\mathrm{2}.\mathrm{3}\left(\frac{\mathrm{2}}{\mathrm{11}}+\mathrm{3}\right) \\ $$

Question Number 57925 Answers: 1 Comments: 1

solve y^(′′) −xy =0 by using integr series.

$${solve}\:{y}^{''} \:−{xy}\:=\mathrm{0}\:\:{by}\:{using}\:{integr}\:{series}. \\ $$

Question Number 57923 Answers: 0 Comments: 3

decompose the fraction F(x) =(x^n /(x^(2n) −1)) inside C(x) and R(x)

$${decompose}\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{{x}^{{n}} }{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:{inside}\:{C}\left({x}\right)\:{and}\:{R}\left({x}\right) \\ $$

Question Number 57922 Answers: 0 Comments: 2

decompose inside C(x) the fraction F(x) =(1/((x^2 +1)^n )) with n integr natural and n≥1

$${decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$${and}\:{n}\geqslant\mathrm{1} \\ $$

Question Number 57915 Answers: 2 Comments: 0

Question Number 57914 Answers: 0 Comments: 0

Question Number 57909 Answers: 2 Comments: 0

n men and n women should be arranged alternately in a row, how many ways can this be done? if the same should be done on a table, how many ways then?

$${n}\:{men}\:{and}\:{n}\:{women}\:{should}\:{be}\:{arranged} \\ $$$${alternately}\:{in}\:{a}\:{row},\:{how}\:{many}\:{ways} \\ $$$${can}\:{this}\:{be}\:{done}?\:{if}\:{the}\:{same}\:{should} \\ $$$${be}\:{done}\:{on}\:{a}\:{table},\:{how}\:{many}\:{ways}\:{then}? \\ $$

Question Number 57902 Answers: 1 Comments: 1

prove that the equation Z^n =1 have exacly n roots given by Z_k =e^(i((2kπ)/n)) k∈[[0,n−1]]

$${prove}\:{that}\:{the}\:{equation}\:{Z}^{{n}} =\mathrm{1}\:\:{have}\:{exacly}\:{n}\:{roots}\:\:{given}\:{by} \\ $$$${Z}_{{k}} ={e}^{{i}\frac{\mathrm{2}{k}\pi}{{n}}} \:\:\:\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right] \\ $$

Question Number 57900 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ ((cos(πxt))/((t^2 +3x^2 )^2 )) dt with x>0 1) find a explicit form for f(x) 2) find the value of ∫_0 ^∞ ((cos(πt))/((t^2 +3)^2 ))dt 3) let U_n =f(n) find nature of Σ U_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{xt}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi{t}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{U}_{{n}} ={f}\left({n}\right)\:\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 57899 Answers: 0 Comments: 2

let f(x) =∫_0 ^(+∞) (dt/((t^2 +x^2 )^3 )) with x>0 1) find a explicit form off (x) 1) calculate ∫_0 ^∞ (dx/((t^2 +3)^3 )) and ∫_0 ^∞ (dt/((t^2 +4)^3 )) 2) find the value of A(θ) =∫_0 ^∞ (dt/((t^2 +sin^2 θ)^3 )) with 0<θ<π.

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\mathrm{3}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{off}\:\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{3}} }\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{sin}^{\mathrm{2}} \theta\right)^{\mathrm{3}} }\:\:{with}\:\mathrm{0}<\theta<\pi. \\ $$

Question Number 57889 Answers: 0 Comments: 0