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) .

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

Pg 1578 Pg 1579 Pg 1580 Pg 1581 Pg 1582 Pg 1583 Pg 1584 Pg 1585 Pg 1586 Pg 1587

Contact: info@tinkutara.com