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Question Number 57434 Answers: 0 Comments: 0

is there a way to find the sum to infinity of a product operator e.g product of 1.2.3.4.5 ... [1, infinity]

$$\mathrm{is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{way}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{infinity}\:\mathrm{of}\:\mathrm{a}\:\mathrm{product}\:\mathrm{operator} \\ $$$$\:\:\mathrm{e}.\mathrm{g}\:\:\:\:\:\mathrm{product}\:\mathrm{of}\:\:\:\:\:\mathrm{1}.\mathrm{2}.\mathrm{3}.\mathrm{4}.\mathrm{5}\:...\:\:\left[\mathrm{1},\:\mathrm{infinity}\right] \\ $$

Question Number 57433 Answers: 2 Comments: 0

Question Number 57423 Answers: 0 Comments: 0

let A_n =∫_0 ^∞ (dt/((e^t +e^(−t) )^n )) calculate A_n interms of n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({e}^{{t}} \:+\overset{−{t}} {{e}}\right)^{{n}} } \\ $$$${calculate}\:{A}_{{n}} \:{interms}\:{of}\:{n} \\ $$

Question Number 57422 Answers: 0 Comments: 0

let U_n =n ∫_1 ^π ((sinx)/x^n )dx calculate lim_(n→+∞) U_n

$${let}\:{U}_{{n}} ={n}\:\int_{\mathrm{1}} ^{\pi} \:\frac{{sinx}}{{x}^{{n}} }{dx} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

Question Number 57421 Answers: 1 Comments: 0

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

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

Question Number 57420 Answers: 0 Comments: 1

let J(x)=∫_0 ^x (t^2 /((√(t+1)) +(√(t+4))))dt find a explicit form of J(x)

$${let}\:{J}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\sqrt{{t}+\mathrm{1}}\:+\sqrt{{t}+\mathrm{4}}}{dt} \\ $$$${find}\:{a}\:{explicit}\:{form}\:{of}\:{J}\left({x}\right) \\ $$

Question Number 57419 Answers: 0 Comments: 1

find ∫_0 ^1 (x+1) ln(x+(√(1+x^2 )))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({x}+\mathrm{1}\right)\:{ln}\left({x}+\sqrt{\left.\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx}\right. \\ $$

Question Number 57418 Answers: 0 Comments: 1

calculate ∫_(−1) ^4 ((∣x−1∣+∣x−2∣)/(∣x^2 −9∣ +x^2 +16))dx

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{4}} \:\frac{\mid{x}−\mathrm{1}\mid+\mid{x}−\mathrm{2}\mid}{\mid{x}^{\mathrm{2}} −\mathrm{9}\mid\:+{x}^{\mathrm{2}} \:+\mathrm{16}}{dx} \\ $$

Question Number 57417 Answers: 0 Comments: 2

let F(x) =∫_0 ^x ((1+sint)/(2+cost))dt 1) find a explicite form of f(x) 2) calculate ∫_0 ^π ((1+sint)/(2+cost))dt

$${let}\:{F}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{\mathrm{1}+{sint}}{\mathrm{2}+{cost}}{dt} \\ $$

Question Number 57416 Answers: 0 Comments: 1

let f(x)=∫_(2x) ^(4x) (dt/(t^2 −2t +3)) 1)find f(x) 2) calculate lim_(x→0) f(x) and lim_(x→+∞) f(x)

$${let}\:{f}\left({x}\right)=\int_{\mathrm{2}{x}} ^{\mathrm{4}{x}} \:\:\:\:\frac{{dt}}{{t}^{\mathrm{2}} −\mathrm{2}{t}\:+\mathrm{3}} \\ $$$$\left.\mathrm{1}\right){find}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right) \\ $$

Question Number 57415 Answers: 0 Comments: 1

solve (x−1)y^′ +(1+(√x))y =x e^(−2x)

$${solve}\:\left({x}−\mathrm{1}\right){y}^{'} \:+\left(\mathrm{1}+\sqrt{{x}}\right){y}\:={x}\:{e}^{−\mathrm{2}{x}} \\ $$

Question Number 57414 Answers: 0 Comments: 2

solve y′ =2y^2 +y and y(o)=1

$${solve}\:\:{y}'\:=\mathrm{2}{y}^{\mathrm{2}} \:+{y}\:\:\:{and}\:{y}\left({o}\right)=\mathrm{1} \\ $$

Question Number 57413 Answers: 0 Comments: 0

prove that ln(1+x)>((arctanx)/(1+x)) ∀x>0

$${prove}\:{that}\:{ln}\left(\mathrm{1}+{x}\right)>\frac{{arctanx}}{\mathrm{1}+{x}}\:\:\forall{x}>\mathrm{0} \\ $$

Question Number 57412 Answers: 0 Comments: 1

let u_n =1 +(1/(√2)) +(1/(√3)) +...+(1/(√n)) prove that (u_n ) is divdrgente.

$${let}\:{u}_{{n}} =\mathrm{1}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\:+...+\frac{\mathrm{1}}{\sqrt{{n}}} \\ $$$${prove}\:{that}\:\left({u}_{{n}} \right)\:{is}\:{divdrgente}. \\ $$

Question Number 57411 Answers: 1 Comments: 1

let f(x)=arctan((√x)+(√(x+1))) find f^(−1) (x) .

$${let}\:{f}\left({x}\right)={arctan}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right) \\ $$$${find}\:{f}^{−\mathrm{1}} \left({x}\right)\:. \\ $$

Question Number 57410 Answers: 1 Comments: 1

find lim_(x→1) ((sin(x^5 +x−2))/(x−1))

$${find}\:{lim}_{{x}\rightarrow\mathrm{1}} \:\:\frac{{sin}\left({x}^{\mathrm{5}} \:+{x}−\mathrm{2}\right)}{{x}−\mathrm{1}} \\ $$

Question Number 57409 Answers: 0 Comments: 1

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

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

Question Number 57408 Answers: 0 Comments: 0

let the sequence (a_n ) wich verify a_1 =2 and a_(n+1) =a_n +(√(1+(a_n /n))) prove that ((a_n /n))_(n≥1) is convergente.

$${let}\:{the}\:{sequence}\:\left({a}_{{n}} \right)\:{wich}\:{verify}\:\:\:{a}_{\mathrm{1}} =\mathrm{2}\:\:{and} \\ $$$${a}_{{n}+\mathrm{1}} ={a}_{{n}} \:+\sqrt{\mathrm{1}+\frac{{a}_{{n}} }{{n}}} \\ $$$${prove}\:{that}\:\left(\frac{{a}_{{n}} }{{n}}\right)_{{n}\geqslant\mathrm{1}} \:\:\:{is}\:{convergente}. \\ $$

Question Number 57407 Answers: 0 Comments: 1

let U_0 =cos((π/3)) and U_(n+1) =(√((1+U_n )/2)) find U_n interms of n .

$${let}\:{U}_{\mathrm{0}} ={cos}\left(\frac{\pi}{\mathrm{3}}\right)\:{and}\:{U}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}+{U}_{{n}} }{\mathrm{2}}} \\ $$$${find}\:{U}_{{n}} \:{interms}\:{of}\:{n}\:. \\ $$

Question Number 57406 Answers: 0 Comments: 1

let f(x)=(x/(√(1+x^2 ))) +cos(πx) 1) prove that f(x)=0 have a solurion α inside ]0,1[ 2) use newton method to find a approximate value of α .

$${let}\:{f}\left({x}\right)=\frac{{x}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:+{cos}\left(\pi{x}\right) \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\left({x}\right)=\mathrm{0}\:{have}\:{a}\:{solurion}\:\alpha\:{inside}\:\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right)\:{use}\:{newton}\:{method}\:{to}\:{find}\:{a}\:{approximate}\: \\ $$$${value}\:{of}\:\alpha\:. \\ $$

Question Number 57405 Answers: 1 Comments: 0

calculate lim_(x→0) ((1−cosx.cos(2x)....cos(nx))/x^2 ) with n integr natural not 0.

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−{cosx}.{cos}\left(\mathrm{2}{x}\right)....{cos}\left({nx}\right)}{{x}^{\mathrm{2}} } \\ $$$${with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0}. \\ $$

Question Number 57404 Answers: 1 Comments: 5

find lim_(x→0) ((sin(π(√(cosx))))/x^2 )

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{sin}\left(\pi\sqrt{{cosx}}\right)}{{x}^{\mathrm{2}} } \\ $$

Question Number 57400 Answers: 0 Comments: 1

Question Number 57390 Answers: 0 Comments: 0

a^2 + d^2 − ad = b^2 + c^2 + bc a^2 + b^2 = c^2 + d^2 ((ab + cd)/(ad + bc)) = ?

$${a}^{\mathrm{2}} \:+\:{d}^{\mathrm{2}} \:−\:{ad}\:\:=\:\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:+\:{bc} \\ $$$${a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:\:=\:\:{c}^{\mathrm{2}} \:+\:{d}^{\mathrm{2}} \\ $$$$\frac{{ab}\:+\:{cd}}{{ad}\:+\:{bc}}\:\:=\:\:? \\ $$

Question Number 57389 Answers: 1 Comments: 0

If aεR and the equation : −3{x}^2 +2{x}+a^2 =0 has no integral solution, then all possible value of a lie in the interval : (a)(−1,0)U(0,1) (b)(1,2) (c) (−2,−1) (d)(−∞,−2)U(2,∞)

$${If}\:{a}\epsilon{R}\:{and}\:{the}\:{equation}\:: \\ $$$$−\mathrm{3}\left\{{x}\right\}^{\mathrm{2}} +\mathrm{2}\left\{{x}\right\}+{a}^{\mathrm{2}} =\mathrm{0}\:{has}\:{no}\:{integral} \\ $$$${solution},\:{then}\:{all}\:{possible}\:{value}\:{of}\:{a} \\ $$$${lie}\:{in}\:{the}\:{interval}\:: \\ $$$$\left({a}\right)\left(−\mathrm{1},\mathrm{0}\right)\mathrm{U}\left(\mathrm{0},\mathrm{1}\right)\:\:\:\left({b}\right)\left(\mathrm{1},\mathrm{2}\right) \\ $$$$\left({c}\right)\:\left(−\mathrm{2},−\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\left({d}\right)\left(−\infty,−\mathrm{2}\right)\mathrm{U}\left(\mathrm{2},\infty\right) \\ $$

Question Number 57388 Answers: 0 Comments: 0

Given f(x) = f(x + 2016), ∀x ∈ R If ∫_0 ^3 f(x) = 30, then ∫_3 ^5 f(x + 2016) = ...

$$\mathrm{Given}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{2016}\right),\:\:\forall{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{If}\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:{f}\left({x}\right)\:=\:\mathrm{30},\:\mathrm{then}\:\underset{\mathrm{3}} {\overset{\mathrm{5}} {\int}}\:{f}\left({x}\:+\:\mathrm{2016}\right)\:=\:... \\ $$

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