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let A_n = ∫_0 ^1 x^n e^(−x) dx 1) calculate A_1 and A_2 2) prove that A_(n+1) =(n+1)A_n −(1/e) 3) calculate A_3 , A_4 , and A_5 4) calculate I = ∫_0 ^1 (−x^3 +2x^(2 ) −x)e^(−x) dx

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \:{e}^{−{x}} {dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{\mathrm{1}} \:\:\:{and}\:{A}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\:{A}_{{n}+\mathrm{1}} =\left({n}+\mathrm{1}\right){A}_{{n}} \:−\frac{\mathrm{1}}{{e}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{A}_{\mathrm{3}} \:\:\:,\:{A}_{\mathrm{4}} ,\:{and}\:{A}_{\mathrm{5}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(−{x}^{\mathrm{3}} \:+\mathrm{2}{x}^{\mathrm{2}\:} \:−{x}\right){e}^{−{x}} \:{dx} \\ $$

Question Number 40138 Answers: 0 Comments: 2

let a_k =∫_(−(π/(2 )) +kπ) ^(−(π/2) +(k+1)π) e^(−t) cost dt 1) calculate a_k 2) find lim_(n→+∞) Σ_(k=0) ^n ∣a_k ∣.

$${let}\:\:{a}_{{k}} \:\:\:=\int_{−\frac{\pi}{\mathrm{2}\:}\:+{k}\pi} ^{−\frac{\pi}{\mathrm{2}}\:+\left({k}+\mathrm{1}\right)\pi} \:{e}^{−{t}} \:{cost}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{a}_{{k}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\mid{a}_{{k}} \mid. \\ $$

Question Number 40134 Answers: 0 Comments: 1

calculate ∫_1 ^2 (x^3 /((1+x^4 )^2 ))dx

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$

Question Number 40133 Answers: 0 Comments: 1

find ∫_0 ^1 (dt/((1+t^2 )^2 ))

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

Question Number 40132 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 40131 Answers: 1 Comments: 0

find the value of I = ∫_0 ^1 ((1+x^4 )/(1+x^6 ))dx

$${find}\:{the}\:{value}\:{of}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 40130 Answers: 0 Comments: 1

calculate ∫_0 ^(π/4) cos^4 x sin^2 xdx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{cos}^{\mathrm{4}} {x}\:{sin}^{\mathrm{2}} {xdx} \\ $$

Question Number 40129 Answers: 1 Comments: 0

calculate I = ∫_1 ^2 ((ln(1+t))/t^2 )dt

$${calculate}\:{I}\:=\:\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\frac{{ln}\left(\mathrm{1}+{t}\right)}{{t}^{\mathrm{2}} }{dt} \\ $$

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))}

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

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