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Question Number 50417 Answers: 0 Comments: 1

find ∫_0 ^1 arctan(√(1−(x^2 /2)))dx

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

Question Number 50416 Answers: 1 Comments: 0

calculate ∫_0 ^1 ^3 (√(x^2 (1−x^3 )))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:^{\mathrm{3}} \sqrt{{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{3}} \right)}{dx} \\ $$

Question Number 50415 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) (dt/(1+cosθ cost))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dt}}{\mathrm{1}+{cos}\theta\:{cost}} \\ $$

Question Number 50414 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) ((x sinx cosx)/(tan^2 x +cotan^2 x))dx ctanx =(1/(tanx))

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{x}\:{sinx}\:{cosx}}{{tan}^{\mathrm{2}} {x}\:+{cotan}^{\mathrm{2}} {x}}{dx} \\ $$$${ctanx}\:=\frac{\mathrm{1}}{{tanx}} \\ $$

Question Number 50413 Answers: 0 Comments: 1

let f ∈C^0 (R,R) / ∀ x∈R f(a+b−x)=f(x) 1) find ∫_a ^b xf(x)dx interms of ∫_a ^b f(x)dx 2) calculate ∫_0 ^π ((xdx)/(1+sinx))

$${let}\:{f}\:\in{C}^{\mathrm{0}} \left({R},{R}\right)\:/\:\forall\:{x}\in{R}\:\:{f}\left({a}+{b}−{x}\right)={f}\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:\int_{{a}} ^{{b}} {xf}\left({x}\right){dx}\:{interms}\:{of}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{xdx}}{\mathrm{1}+{sinx}} \\ $$

Question Number 50412 Answers: 0 Comments: 1

1) calculate U_n =∫_0 ^π (dx/(1+cos^2 (nx))) with n from N 2) f continue from [0,π] to R find lim_(n→+∞) ∫_0 ^π ((f(x))/(1+cos^2 (nx)))dx

$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:{with}\:{n}\:{from}\:{N} \\ $$$$\left.\mathrm{2}\right)\:{f}\:{continue}\:{from}\:\left[\mathrm{0},\pi\right]\:{to}\:{R}\:\:{find} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx} \\ $$

Question Number 50411 Answers: 0 Comments: 0

find all function f continues from R to R / ∀(x,h)∈R^2 f(x).f(y)=∫_(x−y) ^(x+y) f(t)dt .

$${find}\:{all}\:{function}\:{f}\:\:{continues}\:{from}\:{R}\:{to}\:{R}\:/ \\ $$$$\forall\left({x},{h}\right)\in{R}^{\mathrm{2}} \:\:\:{f}\left({x}\right).{f}\left({y}\right)=\int_{{x}−{y}} ^{{x}+{y}} \:{f}\left({t}\right){dt}\:. \\ $$

Question Number 50410 Answers: 0 Comments: 0

determine all functions f ∈C^0 (R,R) / ∫_0 ^x f(x)dx =(2/3)xf(x) .

$${determine}\:{all}\:{functions}\:{f}\:\in{C}^{\mathrm{0}} \left({R},{R}\right)\:/ \\ $$$$\int_{\mathrm{0}} ^{{x}} {f}\left({x}\right){dx}\:=\frac{\mathrm{2}}{\mathrm{3}}{xf}\left({x}\right)\:. \\ $$

Question Number 50409 Answers: 0 Comments: 0

find [Σ_(k=1) ^(10^4 ) (1/(√k))]

$${find}\:\left[\sum_{{k}=\mathrm{1}} ^{\mathrm{10}^{\mathrm{4}} } \:\:\frac{\mathrm{1}}{\sqrt{{k}}}\right] \\ $$

Question Number 50408 Answers: 0 Comments: 0

find the value of lim_(n→+∞) Σ_(i=1) ^n Σ_(j=1) ^n (((−1)^(i+j) )/(i+j))

$${find}\:{the}\:{value}\:{of}\:{lim}_{{n}\rightarrow+\infty} \sum_{{i}=\mathrm{1}} ^{{n}} \:\sum_{{j}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{i}+{j}} }{{i}+{j}} \\ $$

Question Number 50407 Answers: 0 Comments: 0

determine f ∈C^0 ([0,1],R) verifying ∫_0 ^1 f(x)dx =(1/3) +∫_0 ^1 (f(x^2 ))^2 dx

$${determine}\:{f}\:\in{C}^{\mathrm{0}} \left(\left[\mathrm{0},\mathrm{1}\right],{R}\right)\:{verifying} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\frac{\mathrm{1}}{\mathrm{3}}\:+\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left({f}\left({x}^{\mathrm{2}} \right)\right)^{\mathrm{2}} {dx} \\ $$

Question Number 50406 Answers: 0 Comments: 2

1) decompose at simple elements U_n = ((n x^(n−1) )/(x^n −1)) 2) calculste ∫_0 ^(2π) (dt/(x−e^(it) ))

$$\left.\mathrm{1}\right)\:{decompose}\:{at}\:{simple}\:{elements} \\ $$$${U}_{{n}} =\:\frac{{n}\:{x}^{{n}−\mathrm{1}} }{{x}^{{n}} −\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{calculste}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{dt}}{{x}−{e}^{{it}} } \\ $$

Question Number 50405 Answers: 1 Comments: 0

let V_n = (1/(2n+1)) +(1/(2n+3)) +...+(1/(4n−1)) determine lim_(n→+∞) V_n

$${let}\:\:{V}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{3}}\:+...+\frac{\mathrm{1}}{\mathrm{4}{n}−\mathrm{1}} \\ $$$${determine}\:{lim}_{{n}\rightarrow+\infty} \:{V}_{{n}} \\ $$

Question Number 50404 Answers: 0 Comments: 0

find lim_(n→+∞) U_n with U_n =(Σ_(k=1) ^(n ) ch((1/(√(k+n)))))−n

$${find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:{with} \\ $$$${U}_{{n}} =\left(\sum_{{k}=\mathrm{1}} ^{{n}\:} \:{ch}\left(\frac{\mathrm{1}}{\sqrt{{k}+{n}}}\right)\right)−{n} \\ $$$$ \\ $$

Question Number 50403 Answers: 0 Comments: 0

study and give the graph for f(x)=x^2 e^((−2x +1)ln(x))

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\: \\ $$$${f}\left({x}\right)={x}^{\mathrm{2}} \:{e}^{\left(−\mathrm{2}{x}\:+\mathrm{1}\right){ln}\left({x}\right)} \\ $$

Question Number 50402 Answers: 0 Comments: 0

study and give the graph for f(x) = e^(−x^2 ) ln(1+3x)

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for} \\ $$$${f}\left({x}\right)\:=\:{e}^{−{x}^{\mathrm{2}} } {ln}\left(\mathrm{1}+\mathrm{3}{x}\right) \\ $$

Question Number 50401 Answers: 1 Comments: 0

study and give the graph for g(x)=(x +(1/x))^x^2

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for}\: \\ $$$${g}\left({x}\right)=\left({x}\:+\frac{\mathrm{1}}{{x}}\right)^{{x}^{\mathrm{2}} } \\ $$

Question Number 50400 Answers: 1 Comments: 0

study and give the graph for f(x) =(x−(1/x))^x

$${study}\:{and}\:{give}\:{the}\:{graph}\:{for} \\ $$$${f}\left({x}\right)\:=\left({x}−\frac{\mathrm{1}}{{x}}\right)^{{x}} \\ $$

Question Number 50399 Answers: 0 Comments: 0

find lim_(x→+∞) (ch(√(x+1))−ch(√x))^(1/(√x))

$${find}\:{lim}_{{x}\rightarrow+\infty} \left({ch}\sqrt{{x}+\mathrm{1}}−{ch}\sqrt{{x}}\right)^{\frac{\mathrm{1}}{\sqrt{{x}}}} \\ $$

Question Number 50398 Answers: 0 Comments: 0

find lim_(t→+∞) ((sh((√(t^2 +t)))−sh(√(t^2 −t)))/((1+(1/t))^t^2 −(t^6 /6)ln^2 (t)))

$${find}\:{lim}_{{t}\rightarrow+\infty} \:\:\:\frac{{sh}\left(\sqrt{{t}^{\mathrm{2}} +{t}}\right)−{sh}\sqrt{{t}^{\mathrm{2}} −{t}}}{\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right)^{{t}^{\mathrm{2}} } \:\:−\frac{{t}^{\mathrm{6}} }{\mathrm{6}}{ln}^{\mathrm{2}} \left({t}\right)} \\ $$

Question Number 50397 Answers: 1 Comments: 0

calculate artan(2)+arctan(5)+arctan(8)

$${calculate}\:{artan}\left(\mathrm{2}\right)+{arctan}\left(\mathrm{5}\right)+{arctan}\left(\mathrm{8}\right) \\ $$

Question Number 50396 Answers: 0 Comments: 0

finf all functions f continues with verfy f(2x+1)=f(x) ∀x∈R

$${finf}\:{all}\:{functions}\:{f}\:{continues}\:{with}\:{verfy} \\ $$$${f}\left(\mathrm{2}{x}+\mathrm{1}\right)={f}\left({x}\right)\:\:\forall{x}\in{R} \\ $$

Question Number 50395 Answers: 0 Comments: 0

let U_n =(e−(1+(1/n))^n )^((√(n^2 +2))−(√(n^2 +1))) calculate lim_(n→+∞) U_n

$${let}\:{U}_{{n}} =\left({e}−\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \right)^{\sqrt{{n}^{\mathrm{2}} \:+\mathrm{2}}−\sqrt{{n}^{\mathrm{2}} \:+\mathrm{1}}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$

Question Number 50394 Answers: 0 Comments: 0

let x∈]0,1[ prove that the equation tan(((πx)/2))=(π/(2nx)) have only one solution x_n 2) study tbe sequence (x_n ) and find a equivalent of x_n

$$\left.{let}\:\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:{prove}\:{that}\:{the}\:{equation}\right. \\ $$$${tan}\left(\frac{\pi{x}}{\mathrm{2}}\right)=\frac{\pi}{\mathrm{2}{nx}}\:{have}\:{only}\:{one}\:{solution}\:{x}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{tbe}\:{sequence}\:\left({x}_{{n}} \right)\:{and}\:{find}\:{a}\:{equivalent}\:{of}\:{x}_{{n}} \\ $$

Question Number 50392 Answers: 0 Comments: 0

let u_0 =5 and ∀n∈N u_(n+1) =u_n +(1/n) prove that 45<u_(1000) <45,1

$${let}\:{u}_{\mathrm{0}} =\mathrm{5}\:{and}\:\forall{n}\in{N}\:\:\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\frac{\mathrm{1}}{{n}} \\ $$$${prove}\:{that}\:\mathrm{45}<{u}_{\mathrm{1000}} <\mathrm{45},\mathrm{1} \\ $$

Question Number 50391 Answers: 0 Comments: 0

let f(t) =(t/(√(1+t))) study the sequence S_n =Σ_(k=1) ^n f((k/n^2 )).

$${let}\:{f}\left({t}\right)\:=\frac{{t}}{\sqrt{\mathrm{1}+{t}}} \\ $$$${study}\:{the}\:{sequence}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:{f}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right). \\ $$

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