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Question Number 73337    Answers: 0   Comments: 3

calculate ∫_0 ^∞ ((cos(artan(2x)))/((3+x^2 )^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({artan}\left(\mathrm{2}{x}\right)\right)}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73336    Answers: 1   Comments: 1

find ∫_0 ^∞ e^(−t) ln(1+e^t )dt

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}} {ln}\left(\mathrm{1}+{e}^{{t}} \right){dt} \\ $$

Question Number 73335    Answers: 1   Comments: 2

eplcit f(x)=∫_0 ^1 ln(x+t+t^2 )dt with x>(1/4) 2)calculate ∫_0 ^1 ln(t^2 +t +(√2))dt

$${eplcit}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}+{t}+{t}^{\mathrm{2}} \right){dt}\:\:\:\:\:\:{with}\:{x}>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({t}^{\mathrm{2}} \:+{t}\:+\sqrt{\mathrm{2}}\right){dt} \\ $$

Question Number 73334    Answers: 1   Comments: 0

calculate lim_(n→+∞) n^2 ( e^(sin((π/n^2 ))) −cos((π/n)))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:{n}^{\mathrm{2}} \left(\:{e}^{{sin}\left(\frac{\pi}{{n}^{\mathrm{2}} }\right)} −{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$

Question Number 73333    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((cos(π +2x^2 ))/((x^2 +4)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi\:+\mathrm{2}{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73332    Answers: 0   Comments: 0

let U_n =Σ_(k=0) ^n (((−1)^k )/(√(2k+1))) determine a equivalent of n when n→+∞

$${let}\:{U}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{\mathrm{2}{k}+\mathrm{1}}}\:\:{determine}\:{a}\:{equivalent}\:{of}\:{n}\:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 73331    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((ln(1+e^(−3x^2 ) ))/(3+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{x}^{\mathrm{2}} } \right)}{\mathrm{3}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 73330    Answers: 1   Comments: 1

find lim_(x→0) ((ln(2−cos(2x)))/(ln(1+xsin(3x))))

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{2}−{cos}\left(\mathrm{2}{x}\right)\right)}{{ln}\left(\mathrm{1}+{xsin}\left(\mathrm{3}{x}\right)\right)} \\ $$

Question Number 73327    Answers: 0   Comments: 1

let w(x)=∫_0 ^∞ ((lnt)/((x^2 +t^2 )^2 ))dt 1) explicit w(x) 2) calculate U_n =∫_0 ^∞ ((lnt)/((n^2 +t^2 )^2 ))dt find lim_(n→+∞) n^4 U_n and determine nature of tbe serie Σ U_n

$${let}\:{w}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{explicit}\:{w}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:{U}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{lnt}}{\left({n}^{\mathrm{2}} \:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{4}} {U}_{{n}} \:\:{and}\:{determine}\:{nature}\:{of}\:{tbe}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 73308    Answers: 0   Comments: 6

what are the solutions of (√(3x^2 +1))=n where n∈N

$${what}\:{are}\:{the}\:{solutions} \\ $$$${of}\:\sqrt{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{1}}={n}\:{where}\:{n}\in\mathbb{N} \\ $$

Question Number 73297    Answers: 2   Comments: 1

Question Number 73295    Answers: 1   Comments: 0

How many solution so that 3n−4, 4n−5, 5n−13 are prime numbers ?

$${How}\:\:{many}\:\:{solution}\:\:{so}\:\:{that} \\ $$$$\mathrm{3}{n}−\mathrm{4},\:\:\mathrm{4}{n}−\mathrm{5},\:\:\mathrm{5}{n}−\mathrm{13} \\ $$$${are}\:\:{prime}\:\:{numbers}\:? \\ $$

Question Number 73293    Answers: 2   Comments: 2

Explicit f(x)= ∫_1 ^∞ ((lnt)/((x^2 +t^2 )^2 )) dt

$${Explicit}\:\:{f}\left({x}\right)=\:\int_{\mathrm{1}} ^{\infty} \:\frac{{lnt}}{\left({x}^{\mathrm{2}} +{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\: \\ $$

Question Number 73279    Answers: 1   Comments: 0

Question Number 73275    Answers: 0   Comments: 2

∫(4/(x^2 (√(4−xδϰ)))) ?

$$ \\ $$$$ \\ $$$$\int\frac{\mathrm{4}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}−{x}\delta\varkappa}}\:\:\:\:? \\ $$$$ \\ $$

Question Number 73274    Answers: 0   Comments: 2

Question Number 73273    Answers: 1   Comments: 0

Question Number 73269    Answers: 0   Comments: 0

Question Number 73260    Answers: 1   Comments: 0

Question Number 73255    Answers: 1   Comments: 0

Question Number 73247    Answers: 1   Comments: 0

Question Number 73243    Answers: 1   Comments: 0

prove that for z ∈C arctanz =(1/(2i))ln(((1+iz)/(1−iz)))

$${prove}\:{that}\:\:{for}\:{z}\:\in{C}\:\:\:{arctanz}\:=\frac{\mathrm{1}}{\mathrm{2}{i}}{ln}\left(\frac{\mathrm{1}+{iz}}{\mathrm{1}−{iz}}\right) \\ $$$$ \\ $$

Question Number 73238    Answers: 1   Comments: 1

let 0<a<1 calculate ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt and ∫_0 ^∞ ((ln^2 (t)t^(a−1) )/(1+t))dt

$${let}\:\mathrm{0}<{a}<\mathrm{1}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}^{\mathrm{2}} \left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt} \\ $$

Question Number 73235    Answers: 0   Comments: 0

Question Number 73261    Answers: 1   Comments: 1

calculate Σ_(n=1) ^∞ (n^4 +2n^2 −3)(x^n /(n!)) in case of convergence.

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left({n}^{\mathrm{4}} +\mathrm{2}{n}^{\mathrm{2}} −\mathrm{3}\right)\frac{{x}^{{n}} }{{n}!}\:{in}\:{case}\:{of}\:{convergence}. \\ $$

Question Number 73231    Answers: 1   Comments: 1

find the sum of Σ_(n=0) ^∞ (n^2 −3n+1)e^(−n)

$${find}\:{the}\:{sum}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{1}\right){e}^{−{n}} \\ $$

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