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Question Number 40128 Answers: 0 Comments: 1

calculate ∫_(−2) ^(−1) (dt/(t(√(1+t^2 )))) .

$${calculate}\:\:\:\int_{−\mathrm{2}} ^{−\mathrm{1}} \:\:\:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}\:. \\ $$

Question Number 40127 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((e^x −1)/(e^x +1))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{e}^{{x}} −\mathrm{1}}{{e}^{{x}} \:+\mathrm{1}}{dx} \\ $$

Question Number 40126 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((tdt)/(1+t^4 ))

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{tdt}}{\mathrm{1}+{t}^{\mathrm{4}} } \\ $$

Question Number 40125 Answers: 0 Comments: 0

let z_0 =0 and ∀n ∈N z_(n+1) =(i/2)z_n +1 1) find z_n at form of sum 2)let W_n =z_n −((1+i)/2) find lim ∣W_n ∣(n→+∞)

$${let}\:{z}_{\mathrm{0}} =\mathrm{0}\:\:{and}\:\:\forall{n}\:\in{N}\:\:\:{z}_{{n}+\mathrm{1}} =\frac{{i}}{\mathrm{2}}{z}_{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:\:{z}_{{n}} {at}\:{form}\:{of}\:{sum} \\ $$$$\left.\mathrm{2}\right){let}\:{W}_{{n}} \:\:={z}_{{n}} \:\:\:\:−\frac{\mathrm{1}+{i}}{\mathrm{2}}\:\:\:{find}\:{lim}\:\mid{W}_{{n}} \mid\left({n}\rightarrow+\infty\right) \\ $$

Question Number 40124 Answers: 0 Comments: 0

let u_0 ≥4 and u_(n+1) =2u_n −3 1) calculate u_n interms of u_0 and n 2)study the convergence of (u^n )

$${let}\:{u}_{\mathrm{0}} \geqslant\mathrm{4}\:\:{and}\:\:{u}_{{n}+\mathrm{1}} =\mathrm{2}{u}_{{n}} −\mathrm{3} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{u}_{{n}} \:{interms}\:{of}\:{u}_{\mathrm{0}} \:{and}\:{n} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{convergence}\:{of}\:\left({u}^{{n}} \right) \\ $$

Question Number 40123 Answers: 1 Comments: 0

study the convergence of v_n = Σ_(k=1) ^n (((−1)^(k+1) )/(2^k +ln(k)))

$${study}\:{the}\:{convergence}\:{of} \\ $$$${v}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{\mathrm{2}^{{k}} \:+{ln}\left({k}\right)} \\ $$

Question Number 40122 Answers: 0 Comments: 0

let u_n = Σ_(k=0) ^n (((−1)^k )/((2k)!)) prove that (u_n ) converges

$${let}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{\left(\mathrm{2}{k}\right)!} \\ $$$${prove}\:{that}\:\left({u}_{{n}} \right)\:{converges} \\ $$

Question Number 40121 Answers: 0 Comments: 0

find assymptotes of f(x)=(√(x^4 −x^2 +x−1)) −(x+1)(√(x^2 +1))

$${find}\:{assymptotes}\:{of}\:{f}\left({x}\right)=\sqrt{{x}^{\mathrm{4}} \:−{x}^{\mathrm{2}} \:+{x}−\mathrm{1}}\:−\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 40120 Answers: 0 Comments: 0

find the value of lim_(x→0) (((1+sinx)^(1/x) −e^(1−(x/2)) )/((1+tanx)^(1/x) −e^(1−(x/2)) ))

$${find}\:{the}\:{value}\:{of}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{\left(\mathrm{1}+{sinx}\right)^{\frac{\mathrm{1}}{{x}}} \:−{e}^{\mathrm{1}−\frac{{x}}{\mathrm{2}}} }{\left(\mathrm{1}+{tanx}\right)^{\frac{\mathrm{1}}{{x}}} \:−{e}^{\mathrm{1}−\frac{{x}}{\mathrm{2}}} } \\ $$

Question Number 40119 Answers: 0 Comments: 0

find lim_(x→+∞) (chx)^(sh(x)) −(shx)^(ch(x))

$${find}\:{lim}_{{x}\rightarrow+\infty} \:\left({chx}\right)^{{sh}\left({x}\right)} \:\:−\left({shx}\right)^{{ch}\left({x}\right)} \\ $$

Question Number 40118 Answers: 0 Comments: 1

calculate lim_(x→0) ((2x)/(ln(((1+x)/(1−x))))) −cosx

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\:\frac{\mathrm{2}{x}}{{ln}\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)}\:−{cosx} \\ $$

Question Number 40117 Answers: 0 Comments: 0

calculate lim_(x→0) (1/(sin^4 x)){ sin((x/(1−x)))−((sinx)/(1−sinx))}

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{1}}{{sin}^{\mathrm{4}} {x}}\left\{\:{sin}\left(\frac{{x}}{\mathrm{1}−{x}}\right)−\frac{{sinx}}{\mathrm{1}−{sinx}}\right\} \\ $$

Question Number 40116 Answers: 0 Comments: 0

let f(x) = ((e^x^2 −1)/x) if x≠0 and f(0)=0 prove that f^(−1) (x)=x−(x^3 /2) +o(x^3 ) (x→0)

$${let}\:{f}\left({x}\right)\:=\:\frac{{e}^{{x}^{\mathrm{2}} } −\mathrm{1}}{{x}}\:\:{if}\:{x}\neq\mathrm{0}\:\:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:\:{f}^{−\mathrm{1}} \left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{2}}\:+{o}\left({x}^{\mathrm{3}} \right)\:\:\:\left({x}\rightarrow\mathrm{0}\right) \\ $$

Question Number 40115 Answers: 0 Comments: 2

let f(x)= ((e^x −1)/x) if x≠0 and f(0)=1 give ∫_0 ^1 f(x)dx at form of serie.

$${let}\:{f}\left({x}\right)=\:\frac{{e}^{{x}} −\mathrm{1}}{{x}}\:\:{if}\:{x}\neq\mathrm{0}\:\:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$${give}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:{at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 40114 Answers: 0 Comments: 1

let g(x)= cos(x+1) developp g at integr serie

$${let}\:{g}\left({x}\right)=\:{cos}\left({x}+\mathrm{1}\right) \\ $$$${developp}\:{g}\:{at}\:{integr}\:{serie} \\ $$

Question Number 40113 Answers: 0 Comments: 2

let f(x)=ln(2+x) 1) give D_n (0) of f 2) developp f at integr serie

$${let}\:{f}\left({x}\right)={ln}\left(\mathrm{2}+{x}\right) \\ $$$$\left.\mathrm{1}\right)\:{give}\:{D}_{{n}} \left(\mathrm{0}\right)\:{of}\:{f} \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 40112 Answers: 0 Comments: 0

calculate lim_(x→0^+ ) { tan((π/(2+x)))}^x

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:\:\:\left\{\:{tan}\left(\frac{\pi}{\mathrm{2}+{x}}\right)\right\}^{{x}} \\ $$

Question Number 40111 Answers: 0 Comments: 0

find lim _(x→0) ((a^x −b^x )/(e^(ax) −e^(bx) )) with a>0 b>0 and a≠b

$${find}\:\:{lim}\:_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{a}^{{x}} \:−{b}^{{x}} }{{e}^{{ax}} \:−{e}^{{bx}} }\:\:\:{with}\:{a}>\mathrm{0}\:\:\:\:{b}>\mathrm{0}\:\:{and}\:{a}\neq{b} \\ $$

Question Number 40110 Answers: 0 Comments: 0

find lim_(x→+∞) x{(1+a)^(1/x) −a^(1/x) } (a>0)

$${find}\:\:{lim}_{{x}\rightarrow+\infty} {x}\left\{\left(\mathrm{1}+{a}\right)^{\frac{\mathrm{1}}{{x}}} \:−{a}^{\frac{\mathrm{1}}{{x}}} \right\}\:\:\:\:\:\left({a}>\mathrm{0}\right) \\ $$

Question Number 40109 Answers: 0 Comments: 0

find lim_(x→+∞) x{(1+a)^(1/x) −a^(1/x) } (a>0)

$${find}\:\:{lim}_{{x}\rightarrow+\infty} {x}\left\{\left(\mathrm{1}+{a}\right)^{\frac{\mathrm{1}}{{x}}} \:−{a}^{\frac{\mathrm{1}}{{x}}} \right\}\:\:\:\:\:\left({a}>\mathrm{0}\right) \\ $$

Question Number 40108 Answers: 0 Comments: 0

calculate lim_(x→(π/2)) (sinx)^(ln∣x−(π/2)∣)

$${calculate}\:\:{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\left({sinx}\right)^{{ln}\mid{x}−\frac{\pi}{\mathrm{2}}\mid} \\ $$

Question Number 40107 Answers: 1 Comments: 0

prove the relations 1) ∀t ∈]0,1] arctan(((√(1−t^2 ))/t))=arccost 2) ∀ t∈[−1,1] 2 arccos(√((1+t)/2)) =arccost

$${prove}\:{the}\:{relations} \\ $$$$\left.\mathrm{1}\left.\right)\left.\:\forall{t}\:\in\right]\mathrm{0},\mathrm{1}\right]\:\:\:{arctan}\left(\frac{\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{{t}}\right)={arccost} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{t}\in\left[−\mathrm{1},\mathrm{1}\right]\:\:\:\:\mathrm{2}\:{arccos}\sqrt{\frac{\mathrm{1}+{t}}{\mathrm{2}}}\:={arccost} \\ $$

Question Number 40106 Answers: 1 Comments: 0

study and give the graph for the function f(x)= (x/(x−1)) e^(1/x)

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\:{the}\:{function} \\ $$$${f}\left({x}\right)=\:\frac{{x}}{{x}−\mathrm{1}}\:{e}^{\frac{\mathrm{1}}{{x}}} \\ $$

Question Number 40105 Answers: 0 Comments: 1

let f(x) = x^n e^(−2nx) with n integr natural calculate f^((n)) (0).

$${let}\:\:{f}\left({x}\right)\:=\:{x}^{{n}} \:{e}^{−\mathrm{2}{nx}} \:\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right). \\ $$

Question Number 40104 Answers: 0 Comments: 0

find number of solution for the equation (e^x /(2(x+1)^2 )) =1 .

$${find}\:{number}\:{of}\:{solution}\:{for}\:{the}\:{equation} \\ $$$$\frac{{e}^{{x}} }{\mathrm{2}\left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:=\mathrm{1}\:. \\ $$

Question Number 40103 Answers: 0 Comments: 0

let g(x)=(√(−x+(√(1+x^2 )))) 1) prove that g is solution for the differencial equation 4(1+x^2 )y^(′′) +4xy^′ −y =0 .prove that g is C^∞ on R 2) determine a relation between g^((n)) (0) and g^((n+2)) (0)

$${let}\:{g}\left({x}\right)=\sqrt{−{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{g}\:{is}\:{solution}\:{for}\:{the}\:{differencial}\:{equation} \\ $$$$\mathrm{4}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}^{''} \:+\mathrm{4}{xy}^{'} \:−{y}\:=\mathrm{0}\:\:\:.{prove}\:{that}\:{g}\:{is}\:{C}^{\infty} {on}\:{R} \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{a}\:{relation}\:{between}\:{g}^{\left({n}\right)} \left(\mathrm{0}\right)\:{and}\:{g}^{\left({n}+\mathrm{2}\right)} \left(\mathrm{0}\right) \\ $$

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