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let A_n = ∫_0 ^1 ((x^(2n+1) ln(x))/(x^2 −1))dx 1) justify the existence of A_n 2)calculate A_(n+1) −A_n 3) prove that x∈]0,1[ ⇒0<((xln(x))/(x^2 −1))<(1/2) 4) find lim_(n→+∞) A_n

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

Question Number 40157 Answers: 1 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 40156 Answers: 1 Comments: 1

find ∫_e^2 ^(+∞) (dt/(tln(t)ln(ln(t)))

$${find}\:\:\:\int_{{e}^{\mathrm{2}} } ^{+\infty} \:\:\:\:\frac{{dt}}{{tln}\left({t}\right){ln}\left({ln}\left({t}\right)\right.} \\ $$

Question Number 40155 Answers: 1 Comments: 1

caoculate ∫_0 ^∞ ((t dt)/((1+t^4 )^2 ))

$${caoculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{t}\:{dt}}{\left(\mathrm{1}+{t}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$

Question Number 40154 Answers: 1 Comments: 1

find the value of ∫_0 ^1 ((ln(t))/((1+t)^2 ))dt

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 40153 Answers: 1 Comments: 1

calculate ∫_1 ^2 ((t−2)/(√(t^2 −1)))dt

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

Question Number 40152 Answers: 1 Comments: 1

let f(x) = ∫_(−1) ^x (e^t /(√(1−e^t )))dt with x<0 1) calculate f(x) 2) find ∫_(−1) ^0 (e^t /(√(1−e^t )))dt

$${let}\:\:{f}\left({x}\right)\:=\:\:\int_{−\mathrm{1}} ^{{x}} \:\:\:\:\frac{{e}^{{t}} }{\sqrt{\mathrm{1}−{e}^{{t}} }}{dt}\:\:\:{with}\:{x}<\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\int_{−\mathrm{1}} ^{\mathrm{0}} \:\:\frac{{e}^{{t}} }{\sqrt{\mathrm{1}−{e}^{{t}} }}{dt} \\ $$

Question Number 40151 Answers: 1 Comments: 1

let F(x) = ∫_0 ^(π/2) cos(xsint)dt 1) prove that ∀u ∈R 1−(u^2 /2) ≤cosu≤1−(u^2 /2) +(u^4 /(24)) 2) prove that (π/2)(1−(x^2 /4))≤F(x)≤ (π/2)(1−(x^2 /4) +(x^4 /(64)))

$${let}\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cos}\left({xsint}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{u}\:\in{R}\:\:\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\leqslant{cosu}\leqslant\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:+\frac{{u}^{\mathrm{4}} }{\mathrm{24}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\right)\leqslant{F}\left({x}\right)\leqslant\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{4}}\:+\frac{{x}^{\mathrm{4}} }{\mathrm{64}}\right) \\ $$

Question Number 40150 Answers: 0 Comments: 1

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

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

Question Number 40149 Answers: 0 Comments: 1

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

$${let}\:{u}_{{n}} =\:\frac{\mathrm{1}}{\sqrt{{n}}}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{4}{k}}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 40148 Answers: 3 Comments: 0

let f(x)= (x^3 /((1+x^2 )^(3/2) )) 1) calculate ∫_0 ^1 f(x)dx 2) let S_n = (1/n^4 ) Σ_(k=1) ^n (k^3 /(√((1+((k/n))^2 )^3 ))) find lim_(n→+∞) S_n

$${let}\:\:{f}\left({x}\right)=\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}_{{n}} =\:\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\:\:\frac{{k}^{\mathrm{3}} }{\sqrt{\left(\mathrm{1}+\left(\frac{{k}}{{n}}\right)^{\mathrm{2}} \right)^{\mathrm{3}} }} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} \\ $$

Question Number 40147 Answers: 0 Comments: 2

calculate ∫_0 ^2 (√(x^3 (2−x)))dx