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Question Number 52449 Answers: 0 Comments: 0

let j=e^((i2π)/3) and P(x)=(1+jx)^n −(1−jx)^n with n integr natural 1) find roots of P(x) 2)factorize P(x) inside C[x] 3) calculate ∫_0 ^1 P(x)dx. 4) decompose inside C(x) the fraction F(x)=(1/(P(x)))

$${let}\:{j}={e}^{\frac{{i}\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{and}\:{P}\left({x}\right)=\left(\mathrm{1}+{jx}\right)^{{n}} −\left(\mathrm{1}−{jx}\right)^{{n}} \:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{of}\:{P}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{P}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {P}\left({x}\right){dx}. \\ $$$$\left.\mathrm{4}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction}\:{F}\left({x}\right)=\frac{\mathrm{1}}{{P}\left({x}\right)} \\ $$

Question Number 52025 Answers: 4 Comments: 8

Question Number 52016 Answers: 1 Comments: 1

Question Number 52012 Answers: 0 Comments: 2

Question Number 52138 Answers: 1 Comments: 0

Question Number 52137 Answers: 0 Comments: 0

Question Number 52007 Answers: 2 Comments: 0

Solve: ((x/4))^(log_5 50x) = x^6

$$\mathrm{Solve}:\:\:\:\:\:\:\:\:\:\:\left(\frac{\boldsymbol{\mathrm{x}}}{\mathrm{4}}\right)^{\boldsymbol{\mathrm{log}}_{\mathrm{5}} \mathrm{50}\boldsymbol{\mathrm{x}}} \:\:\:=\:\:\:\:\boldsymbol{\mathrm{x}}^{\mathrm{6}} \\ $$

Question Number 52006 Answers: 1 Comments: 1

Differentiate sin^(−1) [((ln x)/(cos x))] with respect to tan x^2

$${Differentiate}\:\mathrm{sin}^{−\mathrm{1}} \left[\frac{\mathrm{ln}\:{x}}{\mathrm{cos}\:{x}}\right] \\ $$$${with}\:{respect}\:{to}\:\mathrm{tan}\:{x}^{\mathrm{2}} \\ $$

Question Number 51998 Answers: 0 Comments: 1

let U ={(x,y)∈R^2 / 1≤x^2 +2y^2 ≤3} calculate ∫∫_U ((x−y)/(x^2 +y^2 ))dxdxy

$${let}\:{U}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\mathrm{1}\leqslant{x}^{\mathrm{2}} \:+\mathrm{2}{y}^{\mathrm{2}} \leqslant\mathrm{3}\right\} \\ $$$${calculate}\:\int\int_{{U}} \:\:\:\:\frac{{x}−{y}}{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdxy} \\ $$

Question Number 51997 Answers: 1 Comments: 2

let f(x)=∫_0 ^(π/2) (dt/(1+xsint)) with x>−1 1) calculate f(o) ,f(1) and f(2) 2) give f at form of function

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{xsint}}\:\:{with}\:{x}>−\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({o}\right)\:,{f}\left(\mathrm{1}\right)\:{and}\:{f}\left(\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{give}\:{f}\:{at}\:{form}\:{of}\:{function}\: \\ $$$$ \\ $$

Question Number 51996 Answers: 1 Comments: 1

calculate S_n =Σ_(k=0) ^(n−1) sin((π/(4n)) +((kπ)/(2n)))

$${calculate}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left(\frac{\pi}{\mathrm{4}{n}}\:+\frac{{k}\pi}{\mathrm{2}{n}}\right)\: \\ $$$$ \\ $$

Question Number 51995 Answers: 1 Comments: 0

let f defined on [0,1] by f(0)=0 and f(x)=(1/(2[(1/(2x))]+1)) calculate ∫_0 ^1 f(x)dx

$${let}\:\:{f}\:{defined}\:{on}\:\left[\mathrm{0},\mathrm{1}\right]\:{by}\:\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{and}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{2}\left[\frac{\mathrm{1}}{\mathrm{2}{x}}\right]+\mathrm{1}} \\ $$$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$

Question Number 51994 Answers: 0 Comments: 1

let D_n = {(x,y)∈R^2 /(x,y)∈[(1/n) ,n[ } 1) find the value of ∫∫_D_n e^(−x^2 −y^2 ) dxdy 2) calculate ∫_0 ^(+∞) e^(−x^2 ) dx .

$${let}\:{D}_{{n}} =\:\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} \:\:/\left({x},{y}\right)\in\left[\frac{\mathrm{1}}{{n}}\:,{n}\left[\:\right\}\right.\right. \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\int\int_{{D}_{{n}} } \:\:\:\:\:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } {dxdy} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{+\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:. \\ $$

Question Number 51993 Answers: 0 Comments: 0

find A_n (x)=∫_0 ^1 (1−t^2 )^n cos(tx)dt

$${find}\:{A}_{{n}} \left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {cos}\left({tx}\right){dt} \\ $$

Question Number 51992 Answers: 0 Comments: 0

find lim_(x→0) ∫_x ^(2x) (t/(ln(1+t^2 )))dt

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

Question Number 51991 Answers: 0 Comments: 1

find f(a) =∫ (dx/((√(1+ax^2 ))+(√(1−ax^2 )))) with a>0

$${find}\:{f}\left({a}\right)\:=\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{ax}^{\mathrm{2}} }+\sqrt{\mathrm{1}−{ax}^{\mathrm{2}} }}\:\:{with}\:{a}>\mathrm{0} \\ $$

Question Number 51990 Answers: 1 Comments: 0

calculate ∫_(1/2) ^1 x arctan((√(1−x^2 )))dx

$${calculate}\:\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{1}} \:{x}\:{arctan}\left(\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 51989 Answers: 1 Comments: 0

calculate ∫_(π/4) ^(π/3) ((sinx)/(1+sin^2 x))dx

$${calculate}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\frac{{sinx}}{\mathrm{1}+{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 51988 Answers: 0 Comments: 0

let f(a) =∫ (√(a^2 −x^4 ))dx 1) determine a explicit form of f(a) 2) find ∫ (dx/(√(a^2 −x^4 ))) a>0

$${let}\:{f}\left({a}\right)\:=\int\:\:\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{4}} }{dx} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:\:\:\:\frac{{dx}}{\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{4}} }} \\ $$$${a}>\mathrm{0} \\ $$

Question Number 51987 Answers: 1 Comments: 1

calculate ∫_0 ^(1/2) (√(1−x^4 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \sqrt{\mathrm{1}−{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 51986 Answers: 0 Comments: 0

1) prove that thx =(2/(th(2x))) −(1/(th(x))) 2)simplify S_n =Σ_(k=0) ^n 2^k th(2^k x)

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{thx}\:=\frac{\mathrm{2}}{{th}\left(\mathrm{2}{x}\right)}\:−\frac{\mathrm{1}}{{th}\left({x}\right)} \\ $$$$\left.\mathrm{2}\right){simplify}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\mathrm{2}^{{k}} {th}\left(\mathrm{2}^{{k}} {x}\right) \\ $$

Question Number 51985 Answers: 0 Comments: 1

1) let p integr natural not 0 calculate arctan((p/(p+1)))−arctan(((p−1)/p)) 2)let S_n =Σ_(p=1) ^n arctan((1/(2p^2 ))) find lim_(n→+∞) S_n

$$\left.\mathrm{1}\right)\:{let}\:{p}\:{integr}\:{natural}\:{not}\:\mathrm{0}\:{calculate}\:{arctan}\left(\frac{{p}}{{p}+\mathrm{1}}\right)−{arctan}\left(\frac{{p}−\mathrm{1}}{{p}}\right) \\ $$$$\left.\mathrm{2}\right){let}\:{S}_{{n}} =\sum_{{p}=\mathrm{1}} ^{{n}} \:{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}{p}^{\mathrm{2}} }\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 51984 Answers: 0 Comments: 0

1) prove the convexity of f(x)=ln(1+e^x ) 2) prove that ∀(x_1 ,x_2 ,...,x_n )∈R^n 1+Π_(k=1) ^n (x_k )^(1/n) ≤ Π_(k=1) ^n (x_k +1)^(1/n) 3) prove that 1+(n!)^(1/n) ≤((n+1)!)^(1/n)

$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{convexity}\:{of}\:{f}\left({x}\right)={ln}\left(\mathrm{1}+{e}^{{x}} \right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...,{x}_{{n}} \right)\in{R}^{{n}} \\ $$$$\mathrm{1}+\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}_{{k}} \right)^{\frac{\mathrm{1}}{{n}}} \:\leqslant\:\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}_{{k}} +\mathrm{1}\right)^{\frac{\mathrm{1}}{{n}}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\mathrm{1}+\left({n}!\right)^{\frac{\mathrm{1}}{{n}}} \leqslant\left(\left({n}+\mathrm{1}\right)!\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 51983 Answers: 0 Comments: 0

f is a real function derivable at 0 and f(0)=0 find lim_(n→+∞) Σ_(k=1) ^n f((k/n^2 )) .

$${f}\:{is}\:{a}\:{real}\:{function}\:{derivable}\:{at}\:\mathrm{0}\:{and}\:{f}\left(\mathrm{0}\right)=\mathrm{0}\:{find} \\ $$$${lim}_{{n}\rightarrow+\infty} \sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)\:. \\ $$

Question Number 51982 Answers: 0 Comments: 0

let u_(n+1) =(√(Σ_(k=1) ^n u_k )) with n>0 and u_1 =1 1)calculate u_2 ,u_3 ,u_4 and u_5 2)prove that ∀n≥2 u_(n+) ^2 =u_n ^2 +u_n 3)study the variation of u_n 4)prove that lim_(n→+∞) u_n =+∞ 5)prove that u_(n+1) ∼u_n (n→+∞) 6)let v_n =u_(n+1) −u_n prove that (v_n ) converges and find its limit.

$${let}\:{u}_{{n}+\mathrm{1}} =\sqrt{\sum_{{k}=\mathrm{1}} ^{{n}} \:{u}_{{k}} }\:\:\:\:\:\:\:{with}\:{n}>\mathrm{0}\:\:\:{and}\:{u}_{\mathrm{1}} =\mathrm{1} \\ $$$$\left.\mathrm{1}\right){calculate}\:{u}_{\mathrm{2}} ,{u}_{\mathrm{3}} ,{u}_{\mathrm{4}} {and}\:{u}_{\mathrm{5}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:\forall{n}\geqslant\mathrm{2}\:\:\:\:\:{u}_{{n}+} ^{\mathrm{2}} ={u}_{{n}} ^{\mathrm{2}} \:+{u}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{variation}\:{of}\:{u}_{{n}} \\ $$$$\left.\mathrm{4}\right){prove}\:{that}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} =+\infty \\ $$$$\left.\mathrm{5}\right){prove}\:{that}\:{u}_{{n}+\mathrm{1}} \sim{u}_{{n}} \:\:\left({n}\rightarrow+\infty\right) \\ $$$$\left.\mathrm{6}\right){let}\:{v}_{{n}} ={u}_{{n}+\mathrm{1}} −{u}_{{n}} \:\:{prove}\:{that}\:\left({v}_{{n}} \right)\:{converges}\:{and}\:{find}\:{its}\:{limit}. \\ $$

Question Number 51981 Answers: 0 Comments: 0

let U_n = ((Σ_(k=1) ^n [lnk])/(ln(n!))) determine lim_(n→+∞) U_n

$${let}\:\:{U}_{{n}} =\:\frac{\sum_{{k}=\mathrm{1}} ^{{n}} \left[{lnk}\right]}{{ln}\left({n}!\right)}\:\:{determine}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \\ $$

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