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Question Number 31510 Answers: 0 Comments: 2

find lim_(n→∞) ^n (√(Π_(k=1) ^n (1+(k/n^2 ))))

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

Question Number 31509 Answers: 0 Comments: 1

let S_n =Σ_(k=1) ^n (1/(√(n^2 +2kn))) find lim_(n→∞) S_n .

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{\sqrt{{n}^{\mathrm{2}} \:+\mathrm{2}{kn}}}\:\:{find}\:\:{lim}_{{n}\rightarrow\infty} \:{S}_{{n}} . \\ $$

Question Number 31508 Answers: 0 Comments: 0

let give S_n = Σ_(k=1) ^n (√k) find a simple eqivalent of S_n .

$${let}\:{give}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\sqrt{{k}}\:\:{find}\:{a}\:{simple}\:{eqivalent}\:{of}\:{S}_{{n}} . \\ $$

Question Number 31507 Answers: 0 Comments: 0

g is real function continue let f(x)=∫_0 ^x sin(x−t)g(t)dt 1)prove that f^′ (x)= ∫_0 ^x cos(t−x)g(t)dt 2)prove that f is so<ution of the diff.equa. y^(′′) +y =g(x)

$${g}\:{is}\:{real}\:{function}\:{continue}\:{let} \\ $$$${f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{sin}\left({x}−{t}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{f}^{'} \left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} {cos}\left({t}−{x}\right){g}\left({t}\right){dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{so}<{ution}\:{of}\:{the}\:{diff}.{equa}. \\ $$$${y}^{''} \:+{y}\:={g}\left({x}\right) \\ $$

Question Number 31506 Answers: 0 Comments: 1

let f(x)=∫_x ^(2x) ((sht)/t)dt 1) calculate f^′ (x) 2) find lim_(x→0) f(x) .

$${let}\:{f}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{sht}}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)\:. \\ $$

Question Number 31505 Answers: 0 Comments: 0

find ∫_a ^b ((1−x^2 )/((1+x^2 )(√(1+x^4 ))))dx with a>1 and b>1.

$$\:{find}\:\:\:\:\int_{{a}} ^{{b}} \:\:\:\:\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}{dx}\:\:{with}\:{a}>\mathrm{1}\:{and}\:{b}>\mathrm{1}. \\ $$

Question Number 31504 Answers: 0 Comments: 1

calculate ∫_0 ^1 (dt/(t +(√(1−t^2 )))) .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{{t}\:+\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}\:. \\ $$

Question Number 31503 Answers: 0 Comments: 1

find ∫_2 ^(√5) (dt/(t(√(t^2 −1)))) .

$${find}\:\:\int_{\mathrm{2}} ^{\sqrt{\mathrm{5}}} \:\:\:\:\:\frac{{dt}}{{t}\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}\:. \\ $$

Question Number 31502 Answers: 0 Comments: 3

find f(x)= ∫_0 ^1 ln(1+xt^2 )dt with x>0. 2) give thevalue of ∫_0 ^1 ln(1+t^2 )dt and ∫_0 ^1 ln(1+2t^2 )dt.

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right){dt}\:\:{with}\:{x}>\mathrm{0}. \\ $$$$\left.\mathrm{2}\right)\:{give}\:{thevalue}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right){dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right){dt}. \\ $$

Question Number 31501 Answers: 0 Comments: 1

find ∫_0 ^(π/4) ln(1 +2tanx)dx.

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}\:+\mathrm{2}{tanx}\right){dx}. \\ $$

Question Number 31500 Answers: 0 Comments: 3

let L_n (x)= e^x (e^(−x) x^n )^((n)) 1) prove that L_n is a polynomial 2) find degL_(n ) and the leading coefficient .

$${let}\:{L}_{{n}} \left({x}\right)=\:{e}^{{x}} \:\left({e}^{−{x}} \:{x}^{{n}} \right)^{\left({n}\right)} \: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{L}_{{n}} \:{is}\:{a}\:{polynomial} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{degL}_{{n}\:} {and}\:{the}\:{leading}\:{coefficient}\:. \\ $$

Question Number 31499 Answers: 0 Comments: 1

find the polynial p wich verify p(x)−p^′ (x)=x^n then calculate ∫_0 ^1 p(x)dx.

$${find}\:{the}\:{polynial}\:{p}\:{wich}\:{verify}\:{p}\left({x}\right)−{p}^{'} \left({x}\right)={x}^{{n}} \:{then} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {p}\left({x}\right){dx}. \\ $$

Question Number 31498 Answers: 0 Comments: 0

find tbe value of Π_(k=1) ^n sin(((kπ)/(n+1))).

$${find}\:{tbe}\:{value}\:{of}\:\prod_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\frac{{k}\pi}{{n}+\mathrm{1}}\right). \\ $$

Question Number 31496 Answers: 0 Comments: 1

let a∈]0,π[ and A(x)= x^(2n) −2cos(na)x^n +1 1)factorize inside C[x] A(x) 2) factorize inside R[x] A(x).

$$\left.{let}\:{a}\in\right]\mathrm{0},\pi\left[\:\:\:{and}\:{A}\left({x}\right)=\:{x}^{\mathrm{2}{n}} \:−\mathrm{2}{cos}\left({na}\right){x}^{{n}} \:+\mathrm{1}\right. \\ $$$$\left.\mathrm{1}\right){factorize}\:{inside}\:{C}\left[{x}\right]\:{A}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{R}\left[{x}\right]\:{A}\left({x}\right). \\ $$

Question Number 31495 Answers: 0 Comments: 0

let give p(x)=(x+j)^n −(x−j)^n with j=e^(i((2π)/3)) 1) find roots of p(x) 2) factorize inside C[ x] p(x) 3)factorize inside R[x] p(x).

$${let}\:{give}\:{p}\left({x}\right)=\left({x}+{j}\right)^{{n}} \:−\left({x}−{j}\right)^{{n}} \:{with}\:{j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{inside}\:{C}\left[\:{x}\right]\:{p}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){factorize}\:{inside}\:{R}\left[{x}\right]\:{p}\left({x}\right). \\ $$

Question Number 31494 Answers: 0 Comments: 3

prove that x^2 divide (x+1)^n_ −nx−1 .nintegr.

$${prove}\:{that}\:{x}^{\mathrm{2}} \:{divide}\:\left({x}+\mathrm{1}\right)^{\underset{} {{n}}} \:−{nx}−\mathrm{1}\:.{nintegr}. \\ $$

Question Number 31493 Answers: 0 Comments: 0

if (xcosθ +sint)^n =Q(x^2 +1) +R find tbe polynomialR

$${if}\:\left({xcos}\theta\:+{sint}\right)^{{n}} \:={Q}\left({x}^{\mathrm{2}} +\mathrm{1}\right)\:+{R}\:\:{find}\:{tbe}\:{polynomialR} \\ $$

Question Number 31492 Answers: 0 Comments: 0

find all polynomial p(x) wich verify ∀k∈Z ∫_k ^(k+1) p(x)dx=k+1.

$${find}\:{all}\:{polynomial}\:{p}\left({x}\right)\:{wich}\:{verify}\: \\ $$$$\forall{k}\in{Z}\:\:\:\int_{{k}} ^{{k}+\mathrm{1}} {p}\left({x}\right){dx}={k}+\mathrm{1}. \\ $$

Question Number 31491 Answers: 0 Comments: 0

let p(x)= x^n +a_(n−1) x^(n−1) +.... a_1 x +a_o if ξ is roots of p(x) prove that ∣ξ∣ ≤ 1+max_(0≤i≤n−1) ∣a_i ∣

$${let}\:{p}\left({x}\right)=\:{x}^{{n}} \:+{a}_{{n}−\mathrm{1}} {x}^{{n}−\mathrm{1}} \:+....\:{a}_{\mathrm{1}} {x}\:+{a}_{{o}} \\ $$$${if}\:\:\xi\:\:{is}\:{roots}\:{of}\:{p}\left({x}\right)\:{prove}\:{that}\:\mid\xi\mid\:\leqslant\:\mathrm{1}+{max}_{\mathrm{0}\leqslant{i}\leqslant{n}−\mathrm{1}} \:\mid{a}_{{i}} \mid \\ $$

Question Number 31490 Answers: 0 Comments: 0

simplify p(x)= (1+x^2 )(1+x^4 )....(1+x^(2n) ) with n fromN then find the roots of p(x).

$${simplify}\:{p}\left({x}\right)=\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)....\left(\mathrm{1}+{x}^{\mathrm{2}{n}} \right)\:{with}\:{n}\:{fromN} \\ $$$${then}\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right). \\ $$

Question Number 31485 Answers: 2 Comments: 0

Question Number 31524 Answers: 0 Comments: 1

find lim_(n→∞) (1+sin((1/n)))^n .

$${find}\:{lim}_{{n}\rightarrow\infty} \left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{{n}} . \\ $$

Question Number 31476 Answers: 1 Comments: 0

From a circular disc of radius R a circular hole of radius R/2 is cut out.The centre of the circular hole is at R/2 from the centre of the original disc.Locate the centre of gravity of the resulting flat body. please help me as fast as possible. Thanks!

$${From}\:{a}\:{circular}\:{disc}\:{of}\:{radius}\:{R} \\ $$$${a}\:{circular}\:{hole}\:{of}\:{radius}\:{R}/\mathrm{2}\:{is} \\ $$$${cut}\:{out}.{The}\:{centre}\:{of}\:{the}\:{circular} \\ $$$${hole}\:{is}\:{at}\:{R}/\mathrm{2}\:{from}\:{the}\:{centre}\:{of} \\ $$$${the}\:{original}\:{disc}.{Locate}\:{the}\:{centre} \\ $$$${of}\:{gravity}\:{of}\:{the}\:{resulting}\:{flat} \\ $$$${body}. \\ $$$$ \\ $$$${please}\:{help}\:{me}\:{as}\:{fast}\:{as}\:{possible}. \\ $$$${Thanks}! \\ $$

Question Number 31475 Answers: 0 Comments: 0

Question Number 31474 Answers: 1 Comments: 1

A vacuum chamber has a door with surface dimensions 20cm×20cm.How many 70kg men, each exerting a pull equal to his weight,will it take to pull the door open when the chamber is evacuated to a pressure of 0.1 atm?

$${A}\:{vacuum}\:{chamber}\:{has}\:{a}\:{door} \\ $$$${with}\:{surface}\:{dimensions} \\ $$$$\mathrm{20}{cm}×\mathrm{20}{cm}.{How}\:{many}\:\mathrm{70}{kg}\:{men}, \\ $$$${each}\:{exerting}\:{a}\:{pull}\:{equal}\:{to}\:{his} \\ $$$${weight},{will}\:{it}\:{take}\:{to}\:{pull}\:{the} \\ $$$${door}\:{open}\:{when}\:{the}\:{chamber}\:{is} \\ $$$${evacuated}\:{to}\:{a}\:{pressure}\:{of}\:\mathrm{0}.\mathrm{1}\:{atm}? \\ $$

Question Number 31466 Answers: 0 Comments: 0

let give I_n = ∫_(1/n) ^1 (√(1+t^2 )) dt 1) calculate I_n 2) find lim_(n→∞) I_n .

$${let}\:{give}\:{I}_{{n}} =\:\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:{I}_{{n}} \:\:\:. \\ $$

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