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IntegrationQuestion and Answers: Page 269

Question Number 44573 Answers: 1 Comments: 1

Question Number 44575 Answers: 1 Comments: 3

Question Number 44515 Answers: 1 Comments: 0

let g(x) =∫_0 ^∞ ((t ln(t)dt)/((1+xt)^3 )) with x>0 1) give a explicit form of g(x) 2) calculate ∫_0 ^∞ ((t ln(t))/((1+t)^3 ))dt 3) calculate ∫_0 ^∞ ((tln(t))/((1+2t)^3 )) dt 4) calculate A(θ) =∫_0 ^∞ ((t ln(t))/((1+t sinθ)^3 ))dt with 0<θ<(π/2)

$${let}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}\:{ln}\left({t}\right){dt}}{\left(\mathrm{1}+{xt}\right)^{\mathrm{3}} }\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{a}\:{explicit}\:{form}\:{of}\:{g}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}\:{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\right)^{\mathrm{3}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{tln}\left({t}\right)}{\left(\mathrm{1}+\mathrm{2}{t}\right)^{\mathrm{3}} }\:{dt} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{A}\left(\theta\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}\:{ln}\left({t}\right)}{\left(\mathrm{1}+{t}\:{sin}\theta\right)^{\mathrm{3}} }{dt}\:\:{with}\:\:\mathrm{0}<\theta<\frac{\pi}{\mathrm{2}} \\ $$

Question Number 44512 Answers: 1 Comments: 1

prove that:−∫2^(ln x) dx = ((x.2^(ln x) )/(ln(xe))) +C

$$\boldsymbol{{prove}}\:\boldsymbol{{that}}:−\int\mathrm{2}^{\boldsymbol{\mathrm{ln}}\:\boldsymbol{\mathrm{x}}} \:\boldsymbol{\mathrm{dx}}\:=\:\frac{\boldsymbol{\mathrm{x}}.\mathrm{2}^{\boldsymbol{\mathrm{ln}}\:\boldsymbol{\mathrm{x}}} }{\boldsymbol{\mathrm{ln}}\left(\boldsymbol{\mathrm{xe}}\right)}\:+\boldsymbol{\mathrm{C}} \\ $$$$ \\ $$

Question Number 44509 Answers: 1 Comments: 1

∫(√(tan x)) dx=?

$$\int\sqrt{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\mathrm{x}}}\:\:\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44508 Answers: 1 Comments: 1

∫(√(sin x ))dx=?

$$\int\sqrt{\boldsymbol{\mathrm{sin}}\:\boldsymbol{\mathrm{x}}\:}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44498 Answers: 0 Comments: 2

Question Number 44476 Answers: 0 Comments: 6

let f(x) =∫_0 ^∞ (dt/(t^2 +2xt−1)) 1)find a explicit form of f(x) 2) cslvulste ∫_0 ^∞ (dt/(t^2 +t−1)) 3)calculate A(θ)=∫_0 ^∞ (dt/(t^2 +2tan(θ)t −1)) 4) calculate g(x)=∫_0 ^∞ ((tdt)/((t^2 +2xt−1)^2 )) 5)find the value of ∫_0 ^∞ ((tdt)/((t^2 +4t−1)^2 ))

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{2}{xt}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{cslvulste}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+{t}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right){calculate}\:{A}\left(\theta\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{2}} \:+\mathrm{2}{tan}\left(\theta\right){t}\:−\mathrm{1}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:+\mathrm{2}{xt}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:+\mathrm{4}{t}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 44475 Answers: 0 Comments: 0

find a and b if ∫_0 ^∞ ((√t) +a(√(t+1))+b(√(t+2)))dt converges and give its value in this case.

$${find}\:{a}\:{and}\:{b}\:\:{if}\:\int_{\mathrm{0}} ^{\infty} \:\left(\sqrt{{t}}\:+{a}\sqrt{{t}+\mathrm{1}}+{b}\sqrt{{t}+\mathrm{2}}\right){dt} \\ $$$${converges}\:{and}\:{give}\:{its}\:{value}\:{in}\:{this}\:{case}. \\ $$

Question Number 44473 Answers: 0 Comments: 1

let A_n =∫_0 ^∞ sin(n[t])e^(−t) dt 2)calculate A_n and lim_(n→+∞) n A_n 3)study the convergence of Σ_n A_n

$${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{sin}\left({n}\left[{t}\right]\right){e}^{−{t}} {dt} \\ $$$$\left.\mathrm{2}\right){calculate}\:{A}_{{n}} \:\:{and}\:{lim}_{{n}\rightarrow+\infty} {n}\:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right){study}\:{the}\:{convergence}\:{of}\:\sum_{{n}} \:{A}_{{n}} \\ $$

Question Number 44472 Answers: 0 Comments: 1

find f(x)=∫_0 ^∞ ((ln(t)dt)/((1+xt)^2 )) withx>0

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){dt}}{\left(\mathrm{1}+{xt}\right)^{\mathrm{2}} }\:{withx}>\mathrm{0} \\ $$

Question Number 44471 Answers: 0 Comments: 2

calculste ∫_0 ^∞ ((ln(x))/((1+x)^2 ))dx

$${calculste}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 44470 Answers: 0 Comments: 0

find ∫_0 ^∞ (dt/(1+t^2 sin^2 t))

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2}} {sin}^{\mathrm{2}} {t}} \\ $$

Question Number 44466 Answers: 0 Comments: 4

let f(x) = ∫_0 ^∞ ((x sinx)/(a^2 +x^4 ))dx with a>0 1) find a explicit form of f(a) 2) find g(a) = ∫_0 ^∞ ((xsinx)/((a^2 +x^4 )^2 ))dx 3)find the value of ∫_0 ^∞ ((x sinx)/(x^4 +1))dx 4) find the value of ∫_0 ^∞ ((xsinx)/((x^4 +1)^2 ))dx .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}\:{sinx}}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}\:{sinx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{xsinx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:. \\ $$$$ \\ $$

Question Number 44441 Answers: 1 Comments: 0