Question and Answers Forum

let I_n (λ) =∫_0 ^π ((vos(nt))/(1−2λcost +λ^2 ))dt 1)calculate I_0 (λ) and I_1 (λ) 2) find relation between I_(n−1) ,I_n and I_(n+1) 3) calculate I_n (λ).

$${let}\:{I}_{{n}} \left(\lambda\right)\:=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{vos}\left({nt}\right)}{\mathrm{1}−\mathrm{2}\lambda{cost}\:+\lambda^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right){calculate}\:{I}_{\mathrm{0}} \left(\lambda\right)\:{and}\:{I}_{\mathrm{1}} \left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{relation}\:{between}\:{I}_{{n}−\mathrm{1}} ,{I}_{{n}} \:{and}\:{I}_{{n}+\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{I}_{{n}} \left(\lambda\right). \\ $$

Question Number 50426 Answers: 2 Comments: 0

study the convergence of U_n =((2/π) ∫_0 ^(π/2) (sinx)^(1/n) )^n

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$${U}_{{n}} =\left(\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({sinx}\right)^{\frac{\mathrm{1}}{{n}}} \right)^{{n}} \\ $$

Question Number 50425 Answers: 0 Comments: 0

find ∫_0 ^∞ ((sin^4 (t))/t^3 ) dt

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}^{\mathrm{4}} \left({t}\right)}{{t}^{\mathrm{3}} }\:{dt} \\ $$

Question Number 50424 Answers: 0 Comments: 0

convergence and calculate ∫_0 ^1 ((ln(t))/((1+t)(√(1−t^2 ))))dt

$${convergence}\:{and}\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)\sqrt{\mathrm{1}−{t}^{\mathrm{2}} }}{dt} \\ $$

Question Number 50422 Answers: 1 Comments: 1

find ∫_0 ^1 ((ln(x))/((√x)(1−x)^(3/2) ))dx

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

Question Number 50421 Answers: 0 Comments: 1

calculate A =∫_0 ^(π/3) (du/((1+cos^2 u)^3 ))

$${calculate}\:{A}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{du}}{\left(\mathrm{1}+{cos}^{\mathrm{2}} {u}\right)^{\mathrm{3}} } \\ $$

Question Number 50420 Answers: 1 Comments: 4

find ∫_0 ^(π/6) cosx ln(cosx)dx

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\:{cosx}\:{ln}\left({cosx}\right){dx} \\ $$

Question Number 50418 Answers: 0 Comments: 1

calculate ∫_0 ^(lln(3)) ((sh^2 (x)dx)/(ch^3 (x)))

$${calculate}\:\int_{\mathrm{0}} ^{{lln}\left(\mathrm{3}\right)} \:\:\frac{{sh}^{\mathrm{2}} \left({x}\right){dx}}{{ch}^{\mathrm{3}} \left({x}\right)} \\ $$

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

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

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