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IntegrationQuestion and Answers: Page 290

Question Number 35603 Answers: 0 Comments: 0

let x ∈ R and {x}=x −[x] prove that ∫_1 ^(+∞) (({x})/x^2 ) dx is convergent and find its value .

$${let}\:{x}\:\in\:{R}\:\:{and}\:\left\{{x}\right\}={x}\:−\left[{x}\right] \\ $$$${prove}\:{that}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{\left\{{x}\right\}}{{x}^{\mathrm{2}} }\:{dx}\:{is}\:{convergent}\:{and}\:{find} \\ $$$${its}\:{value}\:. \\ $$

Question Number 35593 Answers: 0 Comments: 0

let f(a)=∫_0 ^∞ e^(−( t^2 +(a/t^2 ))) dt witha>0 find the value of f(a).

$${let}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left(\:{t}^{\mathrm{2}} \:+\frac{{a}}{{t}^{\mathrm{2}} }\right)} {dt}\:{witha}>\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:{f}\left({a}\right). \\ $$

Question Number 35590 Answers: 0 Comments: 0

find J = ∫_0 ^1 e^(−ax) ln(1+e^(−bx) )dx with a>0 and b>0 .

$${find}\:\:{J}\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−{ax}} {ln}\left(\mathrm{1}+{e}^{−{bx}} \right){dx}\:{with}\:{a}>\mathrm{0}\:{and} \\ $$$${b}>\mathrm{0}\:. \\ $$

Question Number 35589 Answers: 0 Comments: 1

let I = ∫_0 ^∞ e^(−tx) ∣sint∣dt with x>0 find the value of I .

$${let}\:\:\:{I}\:\:=\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{tx}} \:\mid{sint}\mid{dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:{I}\:. \\ $$

Question Number 35588 Answers: 1 Comments: 1

calculate ∫_0 ^(π/3) ((sinx cos(cosx))/(1+2sin(cosx)))dx

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\frac{{sinx}\:{cos}\left({cosx}\right)}{\mathrm{1}+\mathrm{2}{sin}\left({cosx}\right)}{dx} \\ $$

Question Number 35587 Answers: 0 Comments: 0

let f(t) =∫_0 ^1 (e^(−t(1+x^2 )) /(1+x^2 ))dx with t≥0 find a simple form of f(t) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{e}^{−{t}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{with}\:{t}\geqslant\mathrm{0} \\ $$$${find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$

Question Number 35586 Answers: 0 Comments: 0

find the value of f(α) = ∫_0 ^∞ ((arctan(αx))/(1+x^2 ))dx with α from R .

$${find}\:{the}\:{value}\:{of}\: \\ $$$${f}\left(\alpha\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:\:{with}\:\alpha\:{from}\:{R}\:. \\ $$

Question Number 35585 Answers: 0 Comments: 0

let f(x)= ∫_0 ^x sin(cost)dt developp f at integr serie

$${let}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:{sin}\left({cost}\right){dt} \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 35584 Answers: 0 Comments: 0

let f(t) = ∫_0 ^∞ ((arctan(e^(−tx^2 ) ))/x^2 ) dx with t>0 1) study the existence of f(t) 2) calculate f^′ (t)

$${let}\:\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{arctan}\left({e}^{−{tx}^{\mathrm{2}} } \right)}{{x}^{\mathrm{2}} }\:{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:\:{the}\:{existence}\:{of}\:\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$

Question Number 35583 Answers: 0 Comments: 0

find ∫_0 ^(π/6) (√(x−sinx)) dx

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \:\sqrt{{x}−{sinx}}\:{dx} \\ $$

Question Number 35582 Answers: 0 Comments: 0

let g(x)= (1/(2+sinx)) , 2π periodic odd developp f at fourier serie .

$${let}\:{g}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}+{sinx}}\:\:\:\:,\:\mathrm{2}\pi\:{periodic}\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 35581 Answers: 0 Comments: 0

let f(x) = (3/(1+2cosx)) , 2π periodic even developp f at fourier serie.

$${let}\:{f}\left({x}\right)\:=\:\:\frac{\mathrm{3}}{\mathrm{1}+\mathrm{2}{cosx}}\:\:,\:\mathrm{2}\pi\:{periodic}\:{even} \\ $$$${developp}\:{f}\:\:{at}\:{fourier}\:{serie}. \\ $$$$ \\ $$

Question Number 35580 Answers: 0 Comments: 0

if (1/(1+cosx)) = (a_0 /2) +Σ_(n=1) ^∞ a_n cos(nx) calculate a_0 and a_n

$${if}\:\:\:\frac{\mathrm{1}}{\mathrm{1}+{cosx}}\:=\:\frac{{a}_{\mathrm{0}} }{\mathrm{2}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:{a}_{{n}} \:{cos}\left({nx}\right)\:\:{calculate}\:{a}_{\mathrm{0}} \\ $$$${and}\:{a}_{{n}} \\ $$

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Question Number 35471 Answers: 0 Comments: 0

∫_a ^b f(x)dx=area under the curve but say why what is the meaning of ∫ ←this sign

$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}={area}\:{under}\:{the}\:{curve}\:{but}\:{say}\:{why} \\ $$$${what}\:{is}\:{the}\:{meaning}\:{of}\:\int\:\leftarrow{this}\:{sign} \\ $$

Question Number 35456 Answers: 2 Comments: 0

∫(dx/(x(x^(2018) +1)))

$$\int\frac{{dx}}{{x}\left({x}^{\mathrm{2018}} +\mathrm{1}\right)} \\ $$

Question Number 35440 Answers: 1 Comments: 2

find the value of ∫_0 ^∞ (dx/((2x^2 +1)^2 ))

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{2}{x}^{\mathrm{2}} \:\:+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

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