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Question Number 40984 Answers: 0 Comments: 1

let f(x) =∫_0 ^∞ te^(−t^2 ) arctan(xt)dt 1) find a simple form of f(x) 2) calculate ∫_0 ^∞ te^(−t^2 ) arctantdt and ∫_0 ^∞ t e^(−t^2 ) arctan(2t)dt 3)let u_n =∫_0 ^∞ t e^(−t^2 ) arctan(nt)dt find lim_(n→+∞) u_n study the convergence of Σ u_n

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{te}^{−{t}^{\mathrm{2}} } \:{arctan}\left({xt}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:{te}^{−{t}^{\mathrm{2}} } {arctantdt}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{t}\:{e}^{−{t}^{\mathrm{2}} } \:{arctan}\left(\mathrm{2}{t}\right){dt} \\ $$$$\left.\mathrm{3}\right){let}\:{u}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{t}\:{e}^{−{t}^{\mathrm{2}} } {arctan}\left({nt}\right){dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\Sigma\:{u}_{{n}} \\ $$

Question Number 40975 Answers: 2 Comments: 0

Question Number 40974 Answers: 0 Comments: 0

X = (x_1 , x_2 , ..., x_n ), X∈P^n Does there exist a point in N-dimensions, such that the points and line length (from origin) are prime? That is, Σ_(i=1) ^n x_i ^2 =p^2 , p∈P

$${X}\:=\:\left({x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:...,\:{x}_{{n}} \right),\:\:\:\:{X}\in\mathbb{P}^{{n}} \\ $$$$\: \\ $$$$\mathrm{Does}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{N}-\mathrm{dimensions}, \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{points}\:\mathrm{and}\:\mathrm{line}\:\mathrm{length}\:\left(\mathrm{from}\:\mathrm{origin}\right) \\ $$$$\mathrm{are}\:\mathrm{prime}? \\ $$$$\: \\ $$$$\mathrm{That}\:\mathrm{is}, \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{x}_{{i}} ^{\mathrm{2}} ={p}^{\mathrm{2}} ,\:\:\:{p}\in\mathbb{P} \\ $$

Question Number 40965 Answers: 1 Comments: 0

Question Number 40961 Answers: 1 Comments: 0

Question Number 40960 Answers: 0 Comments: 1

Question Number 40952 Answers: 0 Comments: 1

Question Number 40948 Answers: 2 Comments: 1

Question Number 40939 Answers: 0 Comments: 1

Question Number 40930 Answers: 0 Comments: 1

Question Number 40928 Answers: 1 Comments: 0

Question Number 40920 Answers: 1 Comments: 0

(d^( 2) x/dt^2 )=a−b(1−(l/(√(x^2 +l^2 ))))x Find x(t) if x(0)=x_0 , x′(0)=0 .

$$\frac{{d}^{\:\mathrm{2}} {x}}{{dt}^{\mathrm{2}} }={a}−{b}\left(\mathrm{1}−\frac{{l}}{\sqrt{{x}^{\mathrm{2}} +{l}^{\mathrm{2}} }}\right){x}\: \\ $$$${Find}\:{x}\left({t}\right)\:{if}\:{x}\left(\mathrm{0}\right)={x}_{\mathrm{0}} \:,\:{x}'\left(\mathrm{0}\right)=\mathrm{0}\:. \\ $$

Question Number 40918 Answers: 1 Comments: 0

Question Number 40917 Answers: 1 Comments: 0

Question Number 40911 Answers: 0 Comments: 7

Question Number 40910 Answers: 1 Comments: 0

Question Number 40906 Answers: 2 Comments: 1

f(x) = 5.687cosh((x/(5.687)))−5.687 L=∫_(-11) ^(11) (√(1+[f ′(x)]^2 ))dx

$${f}\left({x}\right)\:=\:\mathrm{5}.\mathrm{687cosh}\left(\frac{{x}}{\mathrm{5}.\mathrm{687}}\right)−\mathrm{5}.\mathrm{687} \\ $$$${L}=\int_{-\mathrm{11}} ^{\mathrm{11}} \sqrt{\mathrm{1}+\left[{f}\:'\left({x}\right)\right]^{\mathrm{2}} }{dx} \\ $$

Question Number 40898 Answers: 2 Comments: 1

let u_n =Σ_(k=1) ^(n−1) ((n−k)/(n−k+1)) find a equivalent of u_n (n→+∞)

$${let}\:{u}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\:\frac{{n}−{k}}{{n}−{k}+\mathrm{1}} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \left({n}\rightarrow+\infty\right) \\ $$

Question Number 40897 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (n/((n+1)^2 (n+2)))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}}{\left({n}+\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{2}\right)} \\ $$

Question Number 40896 Answers: 0 Comments: 0

for ∣x∣<1 prove that (1/(√(1−x^2 ))) =Σ_(n=0) ^∞ (C_(2n) ^n /4^n ) x^(2n)

$${for}\:\mid{x}\mid<\mathrm{1}\:{prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{C}_{\mathrm{2}{n}} ^{{n}} }{\mathrm{4}^{{n}} }\:{x}^{\mathrm{2}{n}} \\ $$

Question Number 40895 Answers: 0 Comments: 0

prove that for ∣x∣<1 (1/(√(1+x))) =Σ_(n=0) ^∞ (((−1)^n C_(2n) ^n )/4^k ) x^(2k)

$${prove}\:{that}\:{for}\:\mid{x}\mid<\mathrm{1} \\ $$$$\frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} }{\mathrm{4}^{{k}} }\:{x}^{\mathrm{2}{k}} \\ $$

Question Number 40893 Answers: 0 Comments: 0

let u_k =1−(1−(1/2^k ))^(n−1) 1)prove that Σ u_k converges 2)let f(x)=1−(1−(1/2^x ))^(n−1) with x≥0 prove that ∀p∈N Σ_(k=1) ^(p+1) u_k ≤∫_0 ^(p+1) f(x)dx≤Σ_(k=0) ^p u_k

$${let}\:{u}_{{k}} =\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{k}} }\right)^{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\Sigma\:{u}_{{k}} {converges} \\ $$$$\left.\mathrm{2}\right){let}\:{f}\left({x}\right)=\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\right)^{{n}−\mathrm{1}} \:{with}\:{x}\geqslant\mathrm{0} \\ $$$${prove}\:{that}\:\forall{p}\in{N} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{p}+\mathrm{1}} \:{u}_{{k}} \:\leqslant\int_{\mathrm{0}} ^{{p}+\mathrm{1}} {f}\left({x}\right){dx}\leqslant\sum_{{k}=\mathrm{0}} ^{{p}} \:{u}_{{k}} \\ $$

Question Number 40892 Answers: 0 Comments: 0

let B(x,y) =∫_0 ^1 t^(x−1) (1−t)^(y−1) dt withx>0and y>0 prove that B(x,y)= ((Γ(x).Γ(y))/(Γ(x+y)))

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$${withx}>\mathrm{0}{and}\:{y}>\mathrm{0}\:{prove}\:{that} \\ $$$${B}\left({x},{y}\right)=\:\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$

Question Number 40891 Answers: 0 Comments: 1

let x>0 and y>0 and B(x,y) =∫_0 ^1 t^(x−1) (1−t)^(y−1) dt 1)prove that B(x,y)=B(y,x) 2)B(x+1,y)=(x/y) B(x,y+1) 3)B(x+1,y)=(x/(x+y))B(x,y) 4)B(x,n+1)=((n!)/(x(x+1)....(x+n))) 5)B(n,p) = (1/((n+p−1)C_(n+p−2) ^(p−1) ))

$${let}\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\:{and} \\ $$$${B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$$\left.\mathrm{2}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{y}}\:{B}\left({x},{y}+\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{x}+{y}}{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{4}\right){B}\left({x},{n}+\mathrm{1}\right)=\frac{{n}!}{{x}\left({x}+\mathrm{1}\right)....\left({x}+{n}\right)} \\ $$$$\left.\mathrm{5}\right){B}\left({n},{p}\right)\:=\:\frac{\mathrm{1}}{\left({n}+{p}−\mathrm{1}\right){C}_{{n}+{p}−\mathrm{2}} ^{{p}−\mathrm{1}} } \\ $$

Question Number 40890 Answers: 0 Comments: 2

1)calculate ∫_(1/(n+1)) ^(1/n) [(1/t)−[(1/t)]]dt 2)prove that ∫_0 ^1 [(1/t)−[(1/t)]]dt=1−γ γ is constant number of euler

$$\left.\mathrm{1}\right){calculate}\:\int_{\frac{\mathrm{1}}{{n}+\mathrm{1}}} ^{\frac{\mathrm{1}}{{n}}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt}=\mathrm{1}−\gamma \\ $$$$\gamma\:{is}\:{constant}\:{number}\:{of}\:{euler} \\ $$

Question Number 40889 Answers: 1 Comments: 0

prove?that ∫_0 ^1 ((1−(1−t)^n )/t)dt =Σ_(k=1) ^n (1/k)

$${prove}?{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\left(\mathrm{1}−{t}\right)^{{n}} }{{t}}{dt}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

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