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IntegrationQuestion and Answers: Page 268

Question Number 44777 Answers: 0 Comments: 0

Question Number 44712 Answers: 3 Comments: 0

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Question Number 44697 Answers: 0 Comments: 0

prove that:− ∫_0 ^∞ (t^(a−1) /(1+t))dt = (𝛑/(sin(𝛑a)))

$$\boldsymbol{{prove}}\:\boldsymbol{{that}}:− \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{\boldsymbol{{t}}^{\boldsymbol{{a}}−\mathrm{1}} }{\mathrm{1}+\boldsymbol{{t}}}\boldsymbol{{dt}}\:=\:\frac{\boldsymbol{\pi}}{\boldsymbol{{sin}}\left(\boldsymbol{\pi{a}}\right)} \\ $$

Question Number 44706 Answers: 0 Comments: 4

let f_α (x) = ((cos(αx))/(1+x^2 )) 1) calculate f^((n)) (x) and f^((n)) (0) 2) developp f at integr serie 3) give ∫_0 ^x f_α (t) dt at form of serie 4) developp ∫_0 ^∞ f_α (t)dt at integr serie .

$${let}\:{f}_{\alpha} \left({x}\right)\:=\:\frac{{cos}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{give}\:\int_{\mathrm{0}} ^{{x}} \:{f}_{\alpha} \left({t}\right)\:{dt}\:\:{at}\:{form}\:{of}\:{serie}\: \\ $$$$\left.\mathrm{4}\right)\:{developp}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:{f}_{\alpha} \left({t}\right){dt}\:\:{at}\:\:{integr}\:{serie}\:. \\ $$

Question Number 44696 Answers: 1 Comments: 1

∫(1/(1+x^4 ))dx = ?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{4}} }\boldsymbol{\mathrm{dx}}\:=\:? \\ $$

Question Number 44695 Answers: 0 Comments: 2

∫(e^(√(t−1)) /t)dt = ?

$$\int\frac{\boldsymbol{\mathrm{e}}^{\sqrt{\boldsymbol{\mathrm{t}}−\mathrm{1}}} }{\boldsymbol{\mathrm{t}}}\boldsymbol{\mathrm{dt}}\:=\:? \\ $$

Question Number 44639 Answers: 0 Comments: 1

∫(1/(1+(log x)^2 ))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$

Question Number 44654 Answers: 1 Comments: 4

∫(e^x /(1+x^2 ))dx=?

$$\int\frac{\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 44604 Answers: 1 Comments: 1

∫[((log x − 1)/(1+(log x)^2 ))]^2 dx = (x/((log x)^2 +1))+C

$$\int\left[\frac{\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\:\:−\:\:\mathrm{1}}{\mathrm{1}+\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} }\right]^{\mathrm{2}} \boldsymbol{\mathrm{dx}}\:\:=\:\:\frac{\boldsymbol{\mathrm{x}}}{\left(\boldsymbol{\mathrm{log}}\:\boldsymbol{\mathrm{x}}\right)^{\mathrm{2}} +\mathrm{1}}+\boldsymbol{\mathrm{C}} \\ $$

Question Number 44602 Answers: 0 Comments: 0

∫(e^x /(1+x^2 )) dx = ?

$$\int\frac{\boldsymbol{{e}}^{\boldsymbol{\mathrm{x}}} }{\mathrm{1}+\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}}\:=\:\:? \\ $$

Question Number 44587 Answers: 1 Comments: 2

calculate ∫_0 ^∞ (dt/(1+t^(2018) ))

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{2018}} } \\ $$

Question Number 44674 Answers: 0 Comments: 1

solving some integrals we might meet some of the following functions which cannot be solved with elementar knowledge but tables should exist somewhere in the depth of the www... these links might be interesting exponential integral ∫(e^(−x) /x)dx en.wikipedia.org/wiki/Exponential_integral logarithmic integral ∫(dx/(ln x)) en.wikipedia.org/wiki/Logarithmic_integral_function also see en.wikipedia.org/wiki/Polylogarithm trigonometric integrals i.e. ∫((sin x)/x)dx en.wikipedia.org/wiki/Trigonometric_integral

$$\mathrm{solving}\:\mathrm{some}\:\mathrm{integrals}\:\mathrm{we}\:\mathrm{might}\:\mathrm{meet}\:\mathrm{some} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{functions}\:\mathrm{which}\:\mathrm{cannot}\:\mathrm{be} \\ $$$$\mathrm{solved}\:\mathrm{with}\:\mathrm{elementar}\:\mathrm{knowledge}\:\mathrm{but}\:\mathrm{tables} \\ $$$$\mathrm{should}\:\mathrm{exist}\:\mathrm{somewhere}\:\mathrm{in}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{www}... \\ $$$$\mathrm{these}\:\mathrm{links}\:\mathrm{might}\:\mathrm{be}\:\mathrm{interesting} \\ $$$$ \\ $$$$\mathrm{exponential}\:\mathrm{integral} \\ $$$$\int\frac{\mathrm{e}^{−{x}} }{{x}}{dx} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Exponential\_integral} \\ $$$$ \\ $$$$\mathrm{logarithmic}\:\mathrm{integral} \\ $$$$\int\frac{{dx}}{\mathrm{ln}\:{x}} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Logarithmic\_integral\_function} \\ $$$$\mathrm{also}\:\mathrm{see} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Polylogarithm} \\ $$$$ \\ $$$$\mathrm{trigonometric}\:\mathrm{integrals} \\ $$$$\mathrm{i}.\mathrm{e}.\:\int\frac{\mathrm{sin}\:{x}}{{x}}{dx} \\ $$$$\mathrm{en}.\mathrm{wikipedia}.\mathrm{org}/\mathrm{wiki}/\mathrm{Trigonometric\_integral} \\ $$

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))

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

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