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IntegrationQuestion and Answers: Page 304

Question Number 33310 Answers: 0 Comments: 0

let consider the 2π periodic?function f(x) =e^x 1) developp f at fourier serie 2) find the value of Σ_(n=0) ^∞ (((−1)^n )/(n^2 +1))

$${let}\:{consider}\:{the}\:\mathrm{2}\pi\:{periodic}?{function}\:\:{f}\left({x}\right)\:={e}^{{x}} \\ $$$$\left.\mathrm{1}\right)\:{developp}\:{f}\:{at}\:{fourier}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

Question Number 33297 Answers: 0 Comments: 0

find ∫_0 ^(π/2) ln(1+x sinθ)dθ with 0<x<1 2) calculate ∫_0 ^(π/2) ln(1+(1/2)sinθ)dθ

$${find}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left(\mathrm{1}+{x}\:{sin}\theta\right){d}\theta\:\:\:{with}\:\:\mathrm{0}<{x}<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{sin}\theta\right){d}\theta \\ $$

Question Number 33259 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctan(2x))/(a^2 +x^2 )) dx with a≠0

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

Question Number 33258 Answers: 0 Comments: 0

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

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

Question Number 33257 Answers: 0 Comments: 1

let g(x)= (1/(1+x^4 )) 1) find g^((n)) (x) 2) calculate g^((n)) (0) 3) if g(x)=Σ u_n x^n find the sequence u_n

$${let}\:{g}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:{g}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{g}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{if}\:{g}\left({x}\right)=\Sigma\:{u}_{{n}} \:{x}^{{n}} \:\:\:{find}\:{the}\:{sequence}\:{u}_{{n}} \\ $$

Question Number 33222 Answers: 0 Comments: 0

let give n ≥3 integr calculate I_n = ∫_(−∞) ^(+∞) (dx/(1+x +x^2 +....+x^(n−1) ))

$${let}\:{give}\:{n}\:\geqslant\mathrm{3}\:{integr}\:\:{calculate} \\ $$$${I}_{{n}} =\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \:+....+{x}^{{n}−\mathrm{1}} } \\ $$

Question Number 33223 Answers: 0 Comments: 0

let A_n =∫_(−∞) ^(+∞) (e^(iπx) /(1+x+x^2 +...x^(n−1) )) with n≥3 integr find the value of A_n .

$${let}\:\:{A}_{{n}} \:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{i}\pi{x}} }{\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+...{x}^{{n}−\mathrm{1}} }\:\:{with}\:{n}\geqslant\mathrm{3}\:{integr} \\ $$$${find}\:{the}\:{value}\:{of}\:{A}_{{n}} \:. \\ $$

Question Number 33210 Answers: 0 Comments: 0

find lim_(x→+∞) x e^(−x^2 ) ∫^(x−1) _0 e^t^2 dt

$${find}\:\:{lim}_{{x}\rightarrow+\infty} \:\:{x}\:{e}^{−{x}^{\mathrm{2}} } \:\:\:\underset{\mathrm{0}} {\int}^{{x}−\mathrm{1}} \:\:{e}^{{t}^{\mathrm{2}} } \:{dt} \\ $$

Question Number 33204 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cos(ax))/(1+x+x^2 )) dx.

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

Question Number 33232 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((x sin(2x))/((1+4x^2 )^2 )) dx .

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}\:{sin}\left(\mathrm{2}{x}\right)}{\left(\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 33202 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) (dt/((1+t +t^2 )^2 )) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{t}\:+{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:. \\ $$

Question Number 33175 Answers: 0 Comments: 1

find ∫_0 ^1 (dt/((1+t^2 )^2 ))

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$

Question Number 33172 Answers: 0 Comments: 0

find ∫ (dx/(x +(√(x^2 −3x+2)))) .

$${find}\:\:\int\:\:\:\:\frac{{dx}}{{x}\:+\sqrt{{x}^{\mathrm{2}} \:−\mathrm{3}{x}+\mathrm{2}}}\:. \\ $$

Question Number 33170 Answers: 0 Comments: 1

prove that ∫_0 ^∞ ((∣sinx∣)/x) dx is divergent.

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mid{sinx}\mid}{{x}}\:{dx}\:{is}\:{divergent}. \\ $$

Question Number 33169 Answers: 1 Comments: 1

find the value of ∫_0 ^π (dx/(1+2 sin^2 x)) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}\:{sin}^{\mathrm{2}} {x}}\:\:. \\ $$

Question Number 33166 Answers: 0 Comments: 0

find the value of ∫_0 ^1 (dx/(1+x^4 )) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} }\:\:. \\ $$

Question Number 33155 Answers: 0 Comments: 4

Evaluate ∫_(−∞) ^∞ 3x^2 (x^3 + 1)^2 e^(−x^6 − 2x^3 ) dx

$$\mathrm{Evaluate} \\ $$$$\int_{−\infty} ^{\infty} \:\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{3}} \:+\:\mathrm{1}\right)^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{6}} \:−\:\mathrm{2}{x}^{\mathrm{3}} } \:{dx} \\ $$

Question Number 33130 Answers: 0 Comments: 0

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

$${find}\:\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}\:{cos}\theta}{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}}\:{dx}\:. \\ $$

Question Number 33129 Answers: 0 Comments: 2

1)find the value of u_n =∫_(−∞) ^(+∞) ((cos(nx))/(4 +x^2 )) dx 2) find the nature of Σ u_n .

$$\left.\mathrm{1}\right){find}\:{the}\:{value}\:{of}\:\:\:{u}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{4}\:+{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:. \\ $$

Question Number 33128 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )( 1+x^4 ))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 33120 Answers: 1 Comments: 0

let give α>0 find the value of ∫_0 ^1 (dx/(√((1−x)(1+αx)))) .

$${let}\:{give}\:\alpha>\mathrm{0}\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\sqrt{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+\alpha{x}\right)}}\:. \\ $$

Question Number 33119 Answers: 0 Comments: 1

find ∫_0 ^∞ (t^n /(e^t −1)) dt by using ξ(x) for n integr ξ(x)=Σ_(n=1) ^∞ (1/n^x ) with x>1 .

$${find}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{n}} }{{e}^{{t}} \:−\mathrm{1}}\:{dt}\:{by}\:{using}\:\xi\left({x}\right)\:{for}\:{n}\:{integr} \\ $$$$\xi\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{with}\:{x}>\mathrm{1}\:. \\ $$

Question Number 33069 Answers: 0 Comments: 0

by using residus theorem prove that ∫_0 ^∞ (t^(a−1) /(1+t)) dt = (π/(sin(πa))) with 0<a<1 .

$${by}\:\:{using}\:{residus}\:{theorem}\:{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}\:{dt}\:=\:\frac{\pi}{{sin}\left(\pi{a}\right)}\:{with}\:\:\mathrm{0}<{a}<\mathrm{1}\:. \\ $$

Question Number 33028 Answers: 0 Comments: 0

find the value of∫_0 ^∞ (e^(−[t]) /(t+1))dt .

$${find}\:{the}\:{value}\:{of}\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\left[{t}\right]} }{{t}+\mathrm{1}}{dt}\:\:. \\ $$

Question Number 33027 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (x^3 /(1+x^5 ))dx.

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{5}} }{dx}. \\ $$

Question Number 33026 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((1+x^4 )/(1+x^6 )) dx .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{6}} }\:{dx}\:. \\ $$

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