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Question Number 31542 Answers: 0 Comments: 0

let p(x)=(1+x+x^2 )^n −(1−x+x^2 )^n ∈C[x] 1) find the roots of p(x) 2) factorize p(x) inside C[x].

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)^{{n}} \:−\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{{n}} \:\in{C}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right]. \\ $$

Question Number 31538 Answers: 1 Comments: 0

If f(x)= a e^(2x) +b e^x +cx satisfies the condition f(0)= −1, f ′(log 2)=31, ∫_( 0) ^(log 4) (f(x)−cx) dx = ((39)/2), then

$$\mathrm{If}\:{f}\left({x}\right)=\:{a}\:{e}^{\mathrm{2}{x}} +{b}\:{e}^{{x}} +{cx}\:\mathrm{satisfies}\:\mathrm{the} \\ $$$$\mathrm{condition}\:{f}\left(\mathrm{0}\right)=\:−\mathrm{1},\:{f}\:'\left(\mathrm{log}\:\mathrm{2}\right)=\mathrm{31}, \\ $$$$\:\underset{\:\mathrm{0}} {\overset{\mathrm{log}\:\mathrm{4}} {\int}}\left({f}\left({x}\right)−{cx}\right)\:{dx}\:=\:\frac{\mathrm{39}}{\mathrm{2}},\:\mathrm{then} \\ $$

Question Number 31533 Answers: 0 Comments: 0

let put S(x)=Σ_(n=0) ^∞ (((−1)^n )/(n+x)) 1) prove that S is C^1 on]0′+∞[ 2)give the variation of S(x) 3)prove that ∀x>0 S(x+1)+S(x)=(1/x) 4)give a equivalent for S at 0 5)find a equivalent for S at +∞.

$${let}\:{put}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+{x}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{S}\:{is}\:{C}^{\mathrm{1}} \:{on}\right]\mathrm{0}'+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){give}\:{the}\:{variation}\:{of}\:{S}\left({x}\right) \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:\forall{x}>\mathrm{0}\:{S}\left({x}+\mathrm{1}\right)+{S}\left({x}\right)=\frac{\mathrm{1}}{{x}} \\ $$$$\left.\mathrm{4}\right){give}\:{a}\:{equivalent}\:{for}\:{S}\:{at}\:\mathrm{0} \\ $$$$\left.\mathrm{5}\right){find}\:{a}\:{equivalent}\:{for}\:{S}\:{at}\:+\infty. \\ $$

Question Number 31531 Answers: 0 Comments: 0

let give f(x)=(1/x) +Σ_(n=1) ^∞ ((1/(x+n)) +(1/(x−n))) with x∈R−Z 1) prove the existence of f(x) 2)prove that f is 1−periodic 3)prove that f((x/2)) +f(((x+1)/2))=2f(x).

$${let}\:{give}\:{f}\left({x}\right)=\frac{\mathrm{1}}{{x}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\left(\frac{\mathrm{1}}{{x}+{n}}\:+\frac{\mathrm{1}}{{x}−{n}}\right)\:{with}\:{x}\in{R}−{Z} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:\mathrm{1}−{periodic} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{f}\left(\frac{{x}}{\mathrm{2}}\right)\:+{f}\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)=\mathrm{2}{f}\left({x}\right). \\ $$

Question Number 31529 Answers: 0 Comments: 1

find lim_(x→1^− ) lnx.ln(1−x) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{1}^{−} } \:\:\:\:{lnx}.{ln}\left(\mathrm{1}−{x}\right)\:. \\ $$

Question Number 31528 Answers: 0 Comments: 1

find lim_(x→2) ((x^2 −2^x )/(arctanx −artan2)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{2}} \:\:\frac{{x}^{\mathrm{2}} \:\:−\mathrm{2}^{{x}} }{{arctanx}\:−{artan}\mathrm{2}}\:. \\ $$

Question Number 31526 Answers: 0 Comments: 2

1)find lim_(n→∞) ( ((a^(1/n) +b^(1/n) )/2))^n 2) let 0<θ<(π/2) calculate lim_(n→∞) (1/2^n )(^n (√(cosθ)) +^n (√(sinθ)) )^n

$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow\infty} \left(\:\frac{{a}^{\frac{\mathrm{1}}{{n}}} \:+{b}^{\frac{\mathrm{1}}{{n}}} }{\mathrm{2}}\right)^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{let}\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}}\:{calculate}\:{lim}_{{n}\rightarrow\infty} \:\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\left(\:^{{n}} \sqrt{{cos}\theta}\:+^{{n}} \sqrt{{sin}\theta}\:\right)^{{n}} \\ $$

Question Number 31523 Answers: 0 Comments: 0

let give u_n =^(n+1) (√(n+1)) −^n (√n) 1) study the convergence of (u_n ) 2) find nature of serie Σ_(n=1) ^∞ u_n

$${let}\:{give}\:{u}_{{n}} =^{{n}+\mathrm{1}} \sqrt{{n}+\mathrm{1}}\:\:−^{{n}} \sqrt{{n}}\: \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convergence}\:{of}\:\left({u}_{{n}} \right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{serie}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} {u}_{{n}} \\ $$

Question Number 31522 Answers: 0 Comments: 1

let give u_n =(√(ln(n+1)−ln(n))) 1)give a simple eqivalent of u_n (n→∞) 2) deduce the nature of u_n .

$${let}\:{give}\:{u}_{{n}} =\sqrt{{ln}\left({n}+\mathrm{1}\right)−{ln}\left({n}\right)}\: \\ $$$$\left.\mathrm{1}\right){give}\:{a}\:{simple}\:{eqivalent}\:{of}\:{u}_{{n}} \:\left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{deduce}\:{the}\:{nature}\:{of}\:{u}_{{n}} . \\ $$

Question Number 31521 Answers: 0 Comments: 1

study the convergence of u_n =(√(n+1)) −(√(n−1))

$${study}\:{the}\:{convergence}\:{of}\:{u}_{{n}} =\sqrt{{n}+\mathrm{1}}\:−\sqrt{{n}−\mathrm{1}}\: \\ $$

Question Number 31519 Answers: 1 Comments: 0

Question Number 31517 Answers: 0 Comments: 1

find ∫_(−1) ^1 (dx/((√(1+x)) +(√(1−x)))) .

$${find}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:\:. \\ $$

Question Number 31516 Answers: 1 Comments: 1

find ∫ (dx/(x +(√(1+x^2 )))) .

$${find}\:\int\:\:\:\frac{{dx}}{{x}\:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 31515 Answers: 1 Comments: 1

calculate ∫_0 ^1 (dx/(chx)) .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{{chx}}\:. \\ $$

Question Number 31514 Answers: 1 Comments: 0

find ∫_0 ^1 ((arctan(2x))/((1+x)^2 ))dx.

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

Question Number 31513 Answers: 1 Comments: 1

find ∫_0 ^(2π) (dx/(2 +cosx)) .

$${find}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dx}}{\mathrm{2}\:+{cosx}}\:\:. \\ $$

Question Number 31512 Answers: 0 Comments: 1

find lim_(x→∞) ∫_x ^(2x) ((cos((1/t)))/t) dt.

$${find}\:{lim}_{{x}\rightarrow\infty} \:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}\:{dt}. \\ $$

Question Number 31511 Answers: 0 Comments: 0

f is C^2 inside R and a∈R find lim_(h→0) ((fa+h)−2f(a) +f(a−h))/h^2 )

$${f}\:{is}\:{C}^{\mathrm{2}} \:{inside}\:{R}\:{and}\:{a}\in{R}\:{find} \\ $$$${lim}_{{h}\rightarrow\mathrm{0}} \:\frac{\left.{fa}+{h}\right)−\mathrm{2}{f}\left({a}\right)\:+{f}\left({a}−{h}\right)}{{h}^{\mathrm{2}} } \\ $$

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)))) .

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

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