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IntegrationQuestion and Answers: Page 298

Question Number 35681 Answers: 1 Comments: 1

find ∫ arctan(x)dx

$${find}\:\:\int\:{arctan}\left({x}\right){dx} \\ $$

Question Number 35680 Answers: 0 Comments: 0

by using residus theorem calculate W_n =∫_0 ^(π/2) cos^(2n) t dt ( wallis integal) n integr natural .

$${by}\:{using}\:{residus}\:{theorem}\:{calculate} \\ $$$${W}_{{n}} \:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{cos}^{\mathrm{2}{n}} {t}\:{dt}\:\:\left(\:\:{wallis}\:{integal}\right)\:{n}\:{integr} \\ $$$${natural}\:. \\ $$

Question Number 35678 Answers: 0 Comments: 1

let f(t) =∫_0 ^∞ ((e^(−tx^2 ) arctan(x^2 ))/x^2 )dx with t>0 1) study the existencte of f(t) 2)calculate f^′ (t) 3)find a simple form of f(t).

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:{with}\:{t}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{existencte}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right). \\ $$

Question Number 35677 Answers: 0 Comments: 2

find F(x)=∫_0 ^x e^(−2t) cos(t+(π/4))dx.

$${find}\:{F}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{e}^{−\mathrm{2}{t}} {cos}\left({t}+\frac{\pi}{\mathrm{4}}\right){dx}. \\ $$

Question Number 35676 Answers: 0 Comments: 1

find f(x)=∫_0 ^x ch^4 t dt

$${find}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:{ch}^{\mathrm{4}} {t}\:{dt} \\ $$

Question Number 35675 Answers: 0 Comments: 1

calculate ∫_1 ^3 (x/(e^x −1))dx ..

$${calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{x}}{{e}^{{x}} \:−\mathrm{1}}{dx}\:.. \\ $$

Question Number 35635 Answers: 1 Comments: 1

Question Number 35992 Answers: 0 Comments: 1

let f(x)= ((sin(2x))/x) χ_(]−a,a[) (x) with a>0 calculate the fourier trsnsform of f .

$${let}\:{f}\left({x}\right)=\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{x}}\:\chi_{\left.\right]−{a},{a}\left[\right.} \left({x}\right)\:\:{with}\:{a}>\mathrm{0} \\ $$$${calculate}\:{the}\:{fourier}\:{trsnsform}\:{of}\:{f}\:. \\ $$

Question Number 35632 Answers: 0 Comments: 2

let ϕ(x)= (1/(√(a^2 −x^2 ))) if ∣x∣<a and ϕ(x)=0 if ∣x∣≥a find the fourier transform of ϕ .

$${let}\:\varphi\left({x}\right)=\:\frac{\mathrm{1}}{\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }}\:\:{if}\:\mid{x}\mid<{a}\:\:{and}\:\varphi\left({x}\right)=\mathrm{0}\:{if}\:\mid{x}\mid\geqslant{a} \\ $$$${find}\:{the}\:{fourier}\:{transform}\:{of}\:\varphi\:. \\ $$

Question Number 35631 Answers: 0 Comments: 0

let U_n = ∫_0 ^∞ e^(−(t/n)) arctan(t)dt find a equivalent of U_n (n→+∞)

$${let}\:{U}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−\frac{{t}}{{n}}} \:\:{arctan}\left({t}\right){dt} \\ $$$${find}\:{a}\:{equivalent}\:{of}\:{U}_{{n}} \:\:\left({n}\rightarrow+\infty\right) \\ $$

Question Number 35630 Answers: 0 Comments: 5

1) find the value of f(x)=∫_0 ^∞ ((1−cos(xt))/t^2 ) e^(−t) dt 2) calculate ∫_0 ^∞ ((1−cos(t))/t^2 ) e^(−t) dt .

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}−{cos}\left({xt}\right)}{{t}^{\mathrm{2}} }\:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}−{cos}\left({t}\right)}{{t}^{\mathrm{2}} }\:{e}^{−{t}} \:{dt}\:. \\ $$

Question Number 35629 Answers: 0 Comments: 0

let f(x,y) = ∫_x ^y ((ln(t)ln(1−t))/t)dt with 0<x<y<1 give f(x,y) at form of serie .

$${let}\:\:{f}\left({x},{y}\right)\:=\:\int_{{x}} ^{{y}} \:\:\frac{{ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right)}{{t}}{dt}\:\:{with}\:\mathrm{0}<{x}<{y}<\mathrm{1} \\ $$$${give}\:{f}\left({x},{y}\right)\:{at}\:{form}\:{of}\:{serie}\:. \\ $$

Question Number 35628 Answers: 0 Comments: 1

find the value of I =∫_0 ^1 ((ln(t)ln(1−t))/t)dt

$${find}\:{the}\:{value}\:{of}\:\:{I}\:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right)}{{t}}{dt} \\ $$

Question Number 35627 Answers: 0 Comments: 0

study the convergence of I =∫_0 ^∞ (dx/((1+x^2 ∣sinx∣)^(3/2) ))

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$${I}\:\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \mid{sinx}\mid\right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 35625 Answers: 0 Comments: 0

find lim_(ξ→0) ∫_0 ^(π/2) (dx/(√( sin^2 x +ξ cos^2 x)))

$${find}\:{lim}_{\xi\rightarrow\mathrm{0}} \:\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\:\frac{{dx}}{\sqrt{\:{sin}^{\mathrm{2}} {x}\:+\xi\:{cos}^{\mathrm{2}} {x}}} \\ $$

Question Number 35620 Answers: 0 Comments: 1

let f(x) =e^(−x) sinx odd 2π periodic developp f at fourier serie .

$${let}\:\:{f}\left({x}\right)\:={e}^{−{x}} \:{sinx}\:\:\:{odd}\:\mathrm{2}\pi\:{periodic}\: \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 35619 Answers: 0 Comments: 2

let f(x) = x∣x∣ odd 2π periodic developp f at fourier serie .

$${let}\:{f}\left({x}\right)\:=\:{x}\mid{x}\mid\:\:{odd}\:\mathrm{2}\pi\:{periodic} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

Question Number 35618 Answers: 0 Comments: 1

integrate the e.d. y′ +e^(−2x) y = (2x+1)cosx

$${integrate}\:{the}\:{e}.{d}.\:{y}'\:\:+{e}^{−\mathrm{2}{x}} {y}\:=\:\left(\mathrm{2}{x}+\mathrm{1}\right){cosx} \\ $$

Question Number 35617 Answers: 0 Comments: 0

integrate the e.d . y^(′′) +(x−1)y = e^(−x) sinx with y(0) =1

$${integrate}\:{the}\:{e}.{d}\:.\:\:{y}^{''} \:\:+\left({x}−\mathrm{1}\right){y}\:=\:{e}^{−{x}} \:{sinx} \\ $$$${with}\:{y}\left(\mathrm{0}\right)\:=\mathrm{1} \\ $$

Question Number 35616 Answers: 0 Comments: 0

integrate the d.e y^(′′) −2y^′ +y = x^2 ch(x)

$${integrate}\:{the}\:{d}.{e}\:\:{y}^{''} \:−\mathrm{2}{y}^{'} \:+{y}\:=\:{x}^{\mathrm{2}} {ch}\left({x}\right) \\ $$

Question Number 35615 Answers: 0 Comments: 0

let S_n = Σ_(k=0) ^n (1/(3k+1)) calculate S_n interms of H_n with H_n =Σ_(k=1) ^n (1/k)

$${let}\:\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\frac{\mathrm{1}}{\mathrm{3}{k}+\mathrm{1}} \\ $$$${calculate}\:{S}_{{n}} \:\:\:{interms}\:{of}\:{H}_{{n}} \:\:\:{with}\:{H}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{n}} \frac{\mathrm{1}}{{k}} \\ $$

Question Number 35613 Answers: 0 Comments: 0

find I_(a,b) = ∫_(−∞) ^(+∞) (e^x /((1+a e^x )(1+be^x )))dx ..

$${find}\:\:{I}_{{a},{b}} =\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\:\frac{{e}^{{x}} }{\left(\mathrm{1}+{a}\:{e}^{{x}} \right)\left(\mathrm{1}+{be}^{{x}} \right)}{dx}\:.. \\ $$

Question Number 35612 Answers: 0 Comments: 0

calculate I =∫_0 ^∞ (((1+t)^(−(1/4)) −(1+t)^(−(3/4)) )/t)dt

$${calculate}\:{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{1}}{\mathrm{4}}} \:\:\:−\left(\mathrm{1}+{t}\right)^{−\frac{\mathrm{3}}{\mathrm{4}}} }{{t}}{dt}\: \\ $$

Question Number 35611 Answers: 0 Comments: 0

let h(t) = e^(t−e^t ) and for n≥0 we put h_n (t) =nh(nt) calculate ∫_(−∞) ^(+∞) h_n (t)dt .

$${let}\:{h}\left({t}\right)\:=\:{e}^{{t}−{e}^{{t}} } \:\:\:\:{and}\:{for}\:{n}\geqslant\mathrm{0}\:{we}\:{put} \\ $$$${h}_{{n}} \left({t}\right)\:={nh}\left({nt}\right) \\ $$$${calculate}\:\:\int_{−\infty} ^{+\infty} \:{h}_{{n}} \left({t}\right){dt}\:. \\ $$

Question Number 35610 Answers: 0 Comments: 0

let give x∈]0,2π[ and a ∈R,b∈ R prove that ((π−x)/2) = arctan(((sinx)/(1−cosx))) 2) prove that ∣arctan(a)−arctan(b)∣≤∣a−b∣ 3)letθ ∈]0,(π/2)[ , x ∈[θ,2π−θ] , r∈[0,1[ prove that ∣ϕ(x,r) −((π−x)/2)∣≤ ((1−r)/((1−cosθ)^2 ))

$$\left.{let}\:{give}\:{x}\in\right]\mathrm{0},\mathrm{2}\pi\left[\:\:{and}\:{a}\:\in{R},{b}\in\:{R}\right. \\ $$$${prove}\:{that}\:\:\frac{\pi−{x}}{\mathrm{2}}\:=\:{arctan}\left(\frac{{sinx}}{\mathrm{1}−{cosx}}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\mid{arctan}\left({a}\right)−{arctan}\left({b}\right)\mid\leqslant\mid{a}−{b}\mid \\ $$$$\left.\mathrm{3}\left.\right){let}\theta\:\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\left[\:\:,\:{x}\:\in\left[\theta,\mathrm{2}\pi−\theta\right]\:,\:{r}\in\left[\mathrm{0},\mathrm{1}\left[\:{prove}\:{that}\right.\right.\right. \\ $$$$\mid\varphi\left({x},{r}\right)\:−\frac{\pi−{x}}{\mathrm{2}}\mid\leqslant\:\:\frac{\mathrm{1}−{r}}{\left(\mathrm{1}−{cos}\theta\right)^{\mathrm{2}} } \\ $$

Question Number 35609 Answers: 0 Comments: 0

let r ∈[0,1[ and x∈ R and ϕ(x,r) = arctan( ((rsinx)/(1−r cosx))) 1) prove that (∂ϕ/∂x)(x,r) =Σ_(n=1) ^∞ r^n cos(nx) 2)prove that ϕ(x,r) = Σ_(n=1) ^∞ r^n ((sin(nx))/n)

$${let}\:{r}\:\in\left[\mathrm{0},\mathrm{1}\left[\:{and}\:{x}\in\:{R}\:\:{and}\:\right.\right. \\ $$$$\varphi\left({x},{r}\right)\:=\:{arctan}\left(\:\frac{{rsinx}}{\mathrm{1}−{r}\:{cosx}}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\frac{\partial\varphi}{\partial{x}}\left({x},{r}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:{cos}\left({nx}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\varphi\left({x},{r}\right)\:=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{r}^{{n}} \:\:\frac{{sin}\left({nx}\right)}{{n}} \\ $$$$ \\ $$

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