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AllQuestion and Answers: Page 1469

Question Number 62889 Answers: 1 Comments: 0

Question Number 62882 Answers: 0 Comments: 3

let f(x)=ln∣((x−1)/(x+1))∣ 1)determine D_f 2) calculatef^((n)) (x) and f^((n)) (0) 3) developp f at integr serie 4) calculate ∫_(−(1/2)) ^(1/2) f(x)dx .

$${let}\:{f}\left({x}\right)={ln}\mid\frac{{x}−\mathrm{1}}{{x}+\mathrm{1}}\mid \\ $$$$\left.\mathrm{1}\right){determine}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{calculatef}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {f}\left({x}\right){dx}\:. \\ $$

Question Number 62880 Answers: 0 Comments: 1

when f(E^c ) is equal to (f(E))^c

$${when}\:\:{f}\left({E}^{{c}} \right)\:{is}\:{equal}\:{to}\:\left({f}\left({E}\right)\right)^{{c}} \\ $$

Question Number 62879 Answers: 1 Comments: 0

calculate min Σ_(0≤i≤n and 0≤j≤n) (i+j)

$${calculate}\:{min}\:\sum_{\mathrm{0}\leqslant{i}\leqslant{n}\:\:{and}\:\mathrm{0}\leqslant{j}\leqslant{n}} \:\left({i}+{j}\right) \\ $$

Question Number 62878 Answers: 1 Comments: 0

calculate min Σ_(0≤i≤n and 0≤j≤n) i.j

$${calculate}\:{min}\:\sum_{\mathrm{0}\leqslant{i}\leqslant{n}\:{and}\:\mathrm{0}\leqslant{j}\leqslant{n}} \:\:\:\:{i}.{j} \\ $$

Question Number 62877 Answers: 0 Comments: 1

calculate lim_(n→+∞) Σ_(k=0) ^(2n+1) (n/(n^2 +k))

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}+\mathrm{1}} \:\frac{{n}}{{n}^{\mathrm{2}} \:+{k}} \\ $$

Question Number 62874 Answers: 0 Comments: 3

lim_(x→0) ((sin(6x))/(tan(5x))) how to solve this w/o L′hospital′s rule?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sin}\left(\mathrm{6}{x}\right)}{{tan}\left(\mathrm{5}{x}\right)} \\ $$$$ \\ $$$${how}\:\:{to}\:\:{solve}\:\:{this}\:\:{w}/{o}\:\:{L}'{hospital}'{s}\:\:{rule}? \\ $$

Question Number 62861 Answers: 3 Comments: 10

Question Number 62869 Answers: 1 Comments: 0

y(dy/dx) − (y/(dy/dx)) = 2a a is a real number

$${y}\frac{{dy}}{{dx}}\:−\:\frac{{y}}{\frac{{dy}}{{dx}}}\:=\:\mathrm{2}{a} \\ $$$$ \\ $$$${a}\:{is}\:{a}\:{real}\:{number} \\ $$

Question Number 62856 Answers: 0 Comments: 3

let f(λ) =∫_0 ^(+∞) (x^4 /(x^6 +λ^6 )) dx with λ>0 1) calculate f(λ) 2) calculate also g(λ) =∫_0 ^∞ (x^4 /((x^6 +λ^6 )^2 ))dx 3) find the values of ∫_0 ^∞ (x^4 /(x^6 +1)) dx , ∫_0 ^∞ (x^4 /(x^6 +8))dx and ∫_0 ^∞ (x^4 /((x^6 +8)^2 ))dx.

$${let}\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}} }\:{dx}\:\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{f}\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{also}\:{g}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{4}} }{\left({x}^{\mathrm{6}} \:+\lambda^{\mathrm{6}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{6}} \:+\mathrm{1}}\:{dx}\:,\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{6}} \:+\mathrm{8}}{dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{4}} }{\left({x}^{\mathrm{6}} +\mathrm{8}\right)^{\mathrm{2}} }{dx}. \\ $$

Question Number 62855 Answers: 0 Comments: 1

find ∫ ((x^4 /(1+x^6 )))^2 dx 2) calculate ∫_0 ^1 (x^8 /((1+x^6 )^2 ))dx 3) calculate ∫_0 ^(+∞) (x^8 /((1+x^6 )^2 ))dx .

$${find}\:\int\:\:\left(\frac{{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\right)^{\mathrm{2}} \:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{8}} }{\left(\mathrm{1}+{x}^{\mathrm{6}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{x}^{\mathrm{8}} }{\left(\mathrm{1}+{x}^{\mathrm{6}} \right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 62850 Answers: 1 Comments: 1

Question Number 62844 Answers: 1 Comments: 0

Let p(x) = ax^2 + bx + c be such that p(x) takes real values for real values of x and non−real values for non−real values of x . Prove that a = 0 and find all possible values of c.

$${Let}\:{p}\left({x}\right)\:=\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:\:{be}\:{such}\:{that}\:{p}\left({x}\right)\:{takes}\:{real}\:{values} \\ $$$${for}\:{real}\:{values}\:{of}\:{x}\:{and}\:{non}−{real}\:{values}\:{for}\:{non}−{real} \\ $$$${values}\:{of}\:{x}\:.\:{Prove}\:{that}\:{a}\:=\:\mathrm{0}\:{and}\:{find}\:{all} \\ $$$${possible}\:{values}\:{of}\:{c}. \\ $$

Question Number 62839 Answers: 1 Comments: 3

Question Number 62836 Answers: 0 Comments: 0

Question Number 62833 Answers: 0 Comments: 2

∫((cos(x))/x) dx ∫(√(sin(x) )) dx ∫(√(1−k^2 sin^2 (x))) dx k:constant

$$\int\frac{{cos}\left({x}\right)}{{x}}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{{sin}\left({x}\right)\:}\:{dx} \\ $$$$ \\ $$$$\int\sqrt{\mathrm{1}−{k}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({x}\right)}\:{dx}\:\:\:\:\:\:{k}:{constant} \\ $$

Question Number 62828 Answers: 0 Comments: 1

let U_n =∫_0 ^(+∞) ((cos(ch(nx)))/((3+x^2 )^2 ))dx 1) calculate U_n interms of n 2) find lim_(n→+∞) n U_n and lim_(n→+∞) n^2 U_n 3)study the serie Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{cos}\left({ch}\left({nx}\right)\right)}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{U}_{{n}} \:\:\:\:\:{and}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{serie}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 62826 Answers: 0 Comments: 4

Question Number 62821 Answers: 2 Comments: 1

Question Number 62815 Answers: 0 Comments: 2

developp at fourier serie f(x) =cos(tx) ,2π periodic even .

$${developp}\:{at}\:{fourier}\:{serie}\:{f}\left({x}\right)\:={cos}\left({tx}\right)\:\:,\mathrm{2}\pi\:{periodic}\:{even}\:. \\ $$

Question Number 62814 Answers: 0 Comments: 10

Question Number 62813 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt study first the convergence .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({t}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)} {dt} \\ $$$${study}\:{first}\:{the}\:{convergence}\:. \\ $$

Question Number 62812 Answers: 0 Comments: 1

let U_n =∫_0 ^(+∞) ((arctan(nt))/(1+n^2 t^2 ))dt with n natural≥1 1) calculate U_n 2) calculate lim_(n→+∞) n^2 U_n 3) study the convergence of Σ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left({nt}\right)}{\mathrm{1}+{n}^{\mathrm{2}} {t}^{\mathrm{2}} }{dt}\:\:\:\:{with}\:{n}\:{natural}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 62811 Answers: 0 Comments: 2

1) find ∫ ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx 2)calculate ∫_5 ^(+∞) ((2x^2 −1)/((x+1)(x−3)(x^2 −x+2)))dx

$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\:\:\:\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}−\mathrm{3}\right)\left({x}^{\mathrm{2}} −{x}+\mathrm{2}\right)}{dx} \\ $$

Question Number 62809 Answers: 0 Comments: 1

let f(x) = arctan(nx) with n integr natural 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie .

$${let}\:{f}\left({x}\right)\:=\:{arctan}\left({nx}\right)\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 62808 Answers: 0 Comments: 0

f(t) =∫_0 ^(+∞) (e^(−xt) /((x+t)^2 ))dx with t≥0 1) study the set of definition for f(t) 2)study the continuity of f 3)study the derivability of f 4) developp f at integr serie

$${f}\left({t}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{e}^{−{xt}} }{\left({x}+{t}\right)^{\mathrm{2}} }{dx}\:\:\:\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{set}\:{of}\:{definition}\:{for}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{derivability}\:{of}\:{f} \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

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