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Question Number 66379 Answers: 0 Comments: 2

Question Number 66381 Answers: 0 Comments: 0

Question Number 66356 Answers: 1 Comments: 0

Question Number 66355 Answers: 0 Comments: 1

Value of x satiesfied y=((log_4 (x^2 −1))/(4x^2 +2x+1)) negative value is... a. −1<x<(√2) b. −(√2)<x<1 c. −(√2)<x<(√2) d. −(√2)<x<−1 e. x<−2

$${V}\mathrm{alue}\:\mathrm{of}\:{x}\:\mathrm{satiesfied}\:{y}=\frac{\mathrm{log}_{\mathrm{4}} \left({x}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}} \\ $$$${negative}\:{value}\:\mathrm{is}... \\ $$$$\mathrm{a}.\:−\mathrm{1}<{x}<\sqrt{\mathrm{2}} \\ $$$${b}.\:−\sqrt{\mathrm{2}}<{x}<\mathrm{1} \\ $$$${c}.\:−\sqrt{\mathrm{2}}<{x}<\sqrt{\mathrm{2}} \\ $$$${d}.\:−\sqrt{\mathrm{2}}<{x}<−\mathrm{1} \\ $$$${e}.\:{x}<−\mathrm{2} \\ $$

Question Number 66354 Answers: 0 Comments: 1

If 2x+y=8 and (x+y)=(3/2)log_(10) 2.log_8 36 then x^2 +3y=... a. 28 b. 22 c. 20 d. 16 e. 12

$$\mathrm{If}\:\:\mathrm{2}{x}+{y}=\mathrm{8}\:\mathrm{and} \\ $$$$\left({x}+{y}\right)=\frac{\mathrm{3}}{\mathrm{2}}\mathrm{log}_{\mathrm{10}} \:\mathrm{2}.\mathrm{log}_{\mathrm{8}} \mathrm{36} \\ $$$$\mathrm{then}\:\mathrm{x}^{\mathrm{2}} +\mathrm{3y}=... \\ $$$$\mathrm{a}.\:\mathrm{28} \\ $$$$\mathrm{b}.\:\mathrm{22} \\ $$$$\mathrm{c}.\:\mathrm{20} \\ $$$$\mathrm{d}.\:\mathrm{16} \\ $$$$\mathrm{e}.\:\mathrm{12} \\ $$

Question Number 66351 Answers: 0 Comments: 1

let I_n =∫_0 ^∞ (e^(nt) /((1+e^t )^(n+1) ))dt (n from N^★ ) )prove the existence of I_n 2)find lim_(n→+∞) I_n

$${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{{nt}} }{\left(\mathrm{1}+{e}^{{t}} \right)^{{n}+\mathrm{1}} }{dt}\:\:\:\:\:\left({n}\:{from}\:{N}^{\bigstar} \right) \\ $$$$\left.\right){prove}\:{the}\:{existence}\:{of}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:{I}_{{n}} \\ $$

Question Number 66350 Answers: 0 Comments: 1

study the convergence of ∫_0 ^∞ (1−(√(x^n /(2+x^n ))))dx n∈N

$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\left(\mathrm{1}−\sqrt{\frac{{x}^{{n}} }{\mathrm{2}+{x}^{{n}} }}\right){dx}\:\:\:\:{n}\in{N} \\ $$

Question Number 66349 Answers: 0 Comments: 1

study the convergence of ∫_1 ^(+∞) ((arctan(x−1))/((x^2 −1)^(4/3) ))dx

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }{dx} \\ $$

Question Number 66348 Answers: 0 Comments: 0

find nature of ∫_0 ^1 (dx/(e^x −cosx))

$${find}\:{nature}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{{e}^{{x}} −{cosx}} \\ $$

Question Number 66347 Answers: 0 Comments: 0

let I_n =∫_0 ^1 ((x^(2n+1) ln(x))/(x^2 −1))dx 1) prove the existence of I_n 2)calculate I_(n+1) −I_n 3)prove thst x∈]0,1[ ⇒0<((xlnx)/(x^2 −1))<(1/2) 4) find lim_(n→+∞) I_n

$${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{x}^{\mathrm{2}{n}+\mathrm{1}} {ln}\left({x}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{I}_{{n}+\mathrm{1}} −{I}_{{n}} \\ $$$$\left.\mathrm{3}\left.\right){prove}\:{thst}\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\Rightarrow\mathrm{0}<\frac{{xlnx}}{{x}^{\mathrm{2}} −\mathrm{1}}<\frac{\mathrm{1}}{\mathrm{2}}\right. \\ $$$$\left.\mathrm{4}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} \\ $$

Question Number 66346 Answers: 0 Comments: 2

find ∫_0 ^∞ (t^7 /(t^(16) +1))dt

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{\mathrm{7}} }{{t}^{\mathrm{16}} \:+\mathrm{1}}{dt} \\ $$

Question Number 66345 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) (dt/((t^2 −2t +2)^(3/2) ))

$${find}\:{the}\:{value}\:{of}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} −\mathrm{2}{t}\:+\mathrm{2}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 66344 Answers: 0 Comments: 1

let f_n (x)=(1/((1+x^n )^(1+(1/n)) )) defined on [0,1] 1)prove that f_n →^(cs) to a function f on[0,1] 2) calculate I_n =∫_0 ^1 f_n (x)dx

$${let}\:{f}_{{n}} \left({x}\right)=\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{{n}} \right)^{\mathrm{1}+\frac{\mathrm{1}}{{n}}} }\:\:\:{defined}\:{on}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}_{{n}} \rightarrow^{{cs}} \:\:{to}\:{a}\:{function}\:{f}\:{on}\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} {f}_{{n}} \left({x}\right){dx} \\ $$

Question Number 66342 Answers: 0 Comments: 0

let U_n =∫_n ^(n+2) (((t+n)^(1/4) )/t^(1/3) )dt prove that lim_(n→+∞) U_n =0

$${let}\:{U}_{{n}} =\int_{{n}} ^{{n}+\mathrm{2}} \:\:\frac{\left({t}+{n}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{t}^{\frac{\mathrm{1}}{\mathrm{3}}} }{dt}\:\:{prove}\:{that}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} =\mathrm{0} \\ $$

Question Number 66341 Answers: 0 Comments: 1

find lim_(n→+∞) (1/n^4 ) Σ_(k=1) ^n (k^3 /(√((1+((k/n))^2 )^3 )))

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}^{\mathrm{3}} }{\sqrt{\left(\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} \right)^{\mathrm{3}} }} \\ $$

Question Number 66340 Answers: 0 Comments: 1

find ∫_0 ^2 (√(x^3 (2−x)))dx

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} \left(\mathrm{2}−{x}\right)}{dx} \\ $$

Question Number 66339 Answers: 0 Comments: 1

calculate ∫_(1/2) ^1 (dx/((√(4x^2 −1))+(√(4x^2 +1))))

$${calculate}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:\frac{{dx}}{\sqrt{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{1}}+\sqrt{\mathrm{4}{x}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$

Question Number 66338 Answers: 0 Comments: 1

find ∫_(1/2) ^(5/4) ((x^3 dx)/(√(2+x−x^2 )))

$${find}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{5}}{\mathrm{4}}} \:\:\:\frac{{x}^{\mathrm{3}} {dx}}{\sqrt{\mathrm{2}+{x}−{x}^{\mathrm{2}} }} \\ $$

Question Number 66337 Answers: 0 Comments: 2

calculate ∫_(−7) ^(−3) (((x−1)dx)/(√(x^2 +2x−3)))

$${calculate}\:\int_{−\mathrm{7}} ^{−\mathrm{3}} \:\:\frac{\left({x}−\mathrm{1}\right){dx}}{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{2}{x}−\mathrm{3}}} \\ $$

Question Number 66336 Answers: 0 Comments: 0

calculate ∫_0 ^(π/4) ((tanx)/((√2)cosx +2sin^2 x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tanx}}{\sqrt{\mathrm{2}}{cosx}\:+\mathrm{2}{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 66335 Answers: 0 Comments: 1

find ∫_(−(π/6)) ^(π/6) ((1+tanx)/(1+sin(2x)))dx

$${find}\:\:\int_{−\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$

Question Number 66334 Answers: 0 Comments: 2

let I =∫_0 ^(π/4) e^(−2t) cos^4 t dt and J=∫_0 ^(π/4) e^(−2t) sin^4 tdt 1)calculate I+J and I−J 2) find the value of I and J.

$${let}\:{I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{e}^{−\mathrm{2}{t}} \:{cos}^{\mathrm{4}} {t}\:{dt}\:{and}\:{J}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{e}^{−\mathrm{2}{t}} \:{sin}^{\mathrm{4}} {tdt} \\ $$$$\left.\mathrm{1}\right){calculate}\:\:{I}+{J}\:{and}\:{I}−{J} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\:{and}\:{J}. \\ $$

Question Number 66332 Answers: 0 Comments: 0

let A_n =∫_0 ^1 x^n (√(1−x))dx 1)calculate A_0 and A_1 2)prove that ∀n∈N^★ (3+2n)A_n =2nA_(n−1) 3) find A_n interms of n.

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{\mathrm{0}} \:{and}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right){A}_{{n}} =\mathrm{2}{nA}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{A}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 66330 Answers: 0 Comments: 1

let I_n =∫_0 ^1 x^n e^(−x) dx with n integr natural 1) calculate I_0 , I_1 and I_2 2)find arelation between I_n and I_n 3) find I_n interms of n.

$${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \:{e}^{−{x}} \:{dx}\:\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{\mathrm{0}} \:,\:{I}_{\mathrm{1}} \:{and}\:{I}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){find}\:{arelation}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}_{{n}} \:{interms}\:{of}\:{n}. \\ $$

Question Number 66328 Answers: 0 Comments: 1

calculate ∫_0 ^1 (dt/((1+t^2 )^3 ))

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$

Question Number 66326 Answers: 0 Comments: 4

calculate ∫_0 ^1 ((x^4 +1)/(x^6 +1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{4}} \:+\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$

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