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Question Number 35587    Answers: 0   Comments: 0

let f(t) =∫_0 ^1 (e^(−t(1+x^2 )) /(1+x^2 ))dx with t≥0 find a simple form of f(t) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{e}^{−{t}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$

Question Number 35586    Answers: 0   Comments: 0

find the value of f(α) = ∫_0 ^∞ ((arctan(αx))/(1+x^2 ))dx with α from R .

$${find}\:{the}\:{value}\:{of}\: \\ $$$${f}\left(\alpha\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{from}\:{R}\:. \\ $$

Question Number 35585    Answers: 0   Comments: 0

let f(x)= ∫_0 ^x sin(cost)dt developp f at integr serie

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{sin}\left({cost}\right){dt} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 35584    Answers: 0   Comments: 0

let f(t) = ∫_0 ^∞ ((arctan(e^(−tx^2 ) ))/x^2 ) dx with t>0 1) study the existence of f(t) 2) calculate f^′ (t)

$${let}\:\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{arctan}\left({e}^{−{tx}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:\:{the}\:{existence}\:{of}\:\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$

Question Number 35583    Answers: 0   Comments: 0

find ∫_0 ^(π/6) (√(x−sinx)) dx

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\sqrt{{x}−{sinx}}\:{dx} \\ $$

Question Number 35582    Answers: 0   Comments: 0

let g(x)= (1/(2+sinx)) , 2π periodic odd developp f at fourier serie .

$${let}\:{g}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}+{sinx}}\:\:\:\:,\:\mathrm{2}\pi\:{periodic}\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 35581    Answers: 0   Comments: 0

let f(x) = (3/(1+2cosx)) , 2π periodic even developp f at fourier serie.

$${let}\:{f}\left({x}\right)\:=\:\:\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}}\:\:,\:\mathrm{2}\pi\:{periodic}\:{even} \\ $$$${developp}\:{f}\:\:{at}\:{fourier}\:{serie}. \\ $$$$ \\ $$

Question Number 35580    Answers: 0   Comments: 0

if (1/(1+cosx)) = (a_0 /2) +Σ_(n=1) ^∞ a_n cos(nx) calculate a_0 and a_n

$${if}\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} \:{cos}\left({nx}\right)\:\:{calculate}\:{a}_{\mathrm{0}} \\ $$$${and}\:{a}_{{n}} \\ $$

Question Number 35579    Answers: 0   Comments: 0

(u_n ) is a arithmetic sequence with u_0 =1 and u_5 =11 1) find the value of S_n = Σ_(k=0) ^n (1/u_k ^2 ) and lim_(n→+∞) S_n 2) find the value of W_n = Σ_(k=1) ^n (1/(u_(k−1) . u_(k+1) )) and lim_(n→+∞) W_n .

$$\left({u}_{{n}} \right)\:{is}\:{a}\:{arithmetic}\:{sequence}\:{with}\:{u}_{\mathrm{0}} =\mathrm{1}\:{and} \\ $$$${u}_{\mathrm{5}} =\mathrm{11}\:\:\:\:\: \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{the}\:{value}\:{of}\:\:\:{S}_{{n}} \:\:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{u}_{{k}} ^{\mathrm{2}} } \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:{W}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{u}_{{k}−\mathrm{1}} .\:{u}_{{k}+\mathrm{1}} } \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{W}_{{n}} \:. \\ $$

Question Number 35567    Answers: 0   Comments: 2

Question Number 35562    Answers: 0   Comments: 2

Is there a backwards “⇒”?

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{backwards}\:``\Rightarrow''? \\ $$

Question Number 35553    Answers: 0   Comments: 0

Question Number 35551    Answers: 0   Comments: 1

Question Number 35550    Answers: 0   Comments: 0

Question Number 35549    Answers: 0   Comments: 1

Question Number 35548    Answers: 0   Comments: 0

Question Number 35547    Answers: 0   Comments: 0

Question Number 35546    Answers: 0   Comments: 0

Question Number 35545    Answers: 0   Comments: 0

Question Number 35544    Answers: 0   Comments: 1

Question Number 35543    Answers: 0   Comments: 0

Question Number 35541    Answers: 0   Comments: 3

Question Number 35537    Answers: 1   Comments: 0

Question Number 35534    Answers: 1   Comments: 0

If the roots of 9x^2 −2x+7=0 are 2 more than the roots of ax^2 +bx+c=0, then 4a−2b+c can be

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{9}{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{7}=\mathrm{0}\:\mathrm{are}\:\mathrm{2}\:\mathrm{more} \\ $$$$\mathrm{than}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0},\:\mathrm{then} \\ $$$$\mathrm{4}{a}−\mathrm{2}{b}+{c}\:\:\:\mathrm{can}\:\mathrm{be} \\ $$

Question Number 35533    Answers: 1   Comments: 0

If the roots of the equation x^2 −px+q=0 differ by unity, then

$$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{2}} −{px}+{q}=\mathrm{0} \\ $$$$\mathrm{differ}\:\mathrm{by}\:\mathrm{unity},\:\mathrm{then} \\ $$

Question Number 35527    Answers: 0   Comments: 1

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