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IntegrationQuestion and Answers: Page 295

Question Number 34268    Answers: 0   Comments: 0

calculate I = ∫_(−(π/2)) ^(π/2) ln(1+sinx)dx

$${calculate}\:{I}\:=\:\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\mathrm{1}+{sinx}\right){dx} \\ $$

Question Number 34266    Answers: 0   Comments: 0

1) find the relation between ∫_x ^(+∞) e^(−t^2 ) dt and ∫_x ^(+∞) (e^(−t^2 ) /t^2 )dt 2) guive a equivalent to ∫_x ^(+∞) e^(−t^2 ) dt when x→+∞

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{relation}\:{between}\:\int_{{x}} ^{+\infty} \:{e}^{−{t}^{\mathrm{2}} } {dt}\:\:{and} \\ $$$$\int_{{x}} ^{+\infty} \:\:\:\frac{{e}^{−{t}^{\mathrm{2}} } }{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{guive}\:{a}\:{equivalent}\:{to}\:\int_{{x}} ^{+\infty} \:{e}^{−{t}^{\mathrm{2}} } {dt}\:{when}\:{x}\rightarrow+\infty \\ $$

Question Number 34265    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ e^(−2t) sin([t]) dt .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}{t}} \:{sin}\left(\left[{t}\right]\right)\:{dt}\:\:. \\ $$

Question Number 34264    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ e^(−2[t]) sint dt

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\mathrm{2}\left[{t}\right]} {sint}\:{dt} \\ $$

Question Number 34262    Answers: 0   Comments: 0

find the nature of ∫_2 ^(+∞) ((√(1+t^2 +t^4 )) −t ^3 (√(t^3 +at)))dt a∈R .

$${find}\:{the}\:{nature}\:{of}\:\int_{\mathrm{2}} ^{+\infty} \left(\sqrt{\mathrm{1}+{t}^{\mathrm{2}} +{t}^{\mathrm{4}} \:}\:\:−{t}\:\:^{\mathrm{3}} \sqrt{{t}^{\mathrm{3}} +{at}}\right){dt} \\ $$$${a}\in{R}\:. \\ $$

Question Number 34261    Answers: 0   Comments: 0

study the convergence of ∫_0 ^∞ ((t−sint)/t^a )dt with a real.

$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}−{sint}}{{t}^{{a}} }{dt}\:{with}\:{a}\:{real}. \\ $$

Question Number 34260    Answers: 0   Comments: 0

let give a>0 1) find the value of F(a) = ∫_0 ^∞ ((lnt)/(t^2 +a^2 ))dt 2) find the value of G(a)=∫_0 ^∞ ((aln(t))/((t^2 +a^2 )^2 ))dt 3) find the value of ∫_0 ^∞ ((ln(t))/((t^2 +3)^2 ))dt

$${let}\:{give}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:{F}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{lnt}}{{t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{G}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{aln}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left({t}\right)}{\left({t}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 34257    Answers: 0   Comments: 0

find f(x)= ∫_1 ^x (dt/(t(√(1+t^2 )))) 2) calculate I =∫_1 ^(+∞) (dt/(t(√(1+t^2 ))))

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{1}} ^{{x}} \:\:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:=\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$

Question Number 34255    Answers: 0   Comments: 0

find g(x)= ∫_0 ^∞ ((ln(1+xt^2 ))/t^2 ) dt 2) calculate ∫_0 ^∞ ((ln(1+3t^2 ))/t^2 )dt .

$${find}\:\:{g}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+\mathrm{3}{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:. \\ $$

Question Number 34254    Answers: 0   Comments: 0

find I(x)= ∫_0 ^1 ((ln(1+xt^2 ))/t^2 )dt .

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

Question Number 34253    Answers: 0   Comments: 0

let F(x)= ∫_0 ^x ((ln(1+t^2 ))/t^2 )dt 1) calculate F(x) 2) find the value of ∫_0 ^∞ ((ln(1+t^2 ))/t^2 )dt

$$\:{let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt} \\ $$

Question Number 34237    Answers: 0   Comments: 4

find ∫ (dx/(x^2 −a)) with a ∈ C .

$${find}\:\:\int\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:−{a}}\:\:{with}\:{a}\:\in\:{C}\:. \\ $$

Question Number 34229    Answers: 2   Comments: 3

calculate ∫_(−∞) ^∞ ((cos(tx))/(1+x^4 )) dx with t≥0 2) calculate ∫_0 ^∞ (dx/(1+x^4 )) .

$${calculate}\:\int_{−\infty} ^{\infty} \:\:\frac{{cos}\left({tx}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\:\:{with}\:{t}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{4}} }\:. \\ $$

Question Number 34228    Answers: 0   Comments: 1

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

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 34227    Answers: 1   Comments: 2

calculate ∫_0 ^1 arctan(x^2 )dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left({x}^{\mathrm{2}} \right){dx}\: \\ $$

Question Number 34225    Answers: 1   Comments: 0

find ∫ (dx/(1+x^2 +x^4 ))

$${find}\:\int\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} } \\ $$

Question Number 34223    Answers: 0   Comments: 0

find ∫ (dx/(x^(2n) −1)) with n integr natural and n≥1 .

$${find}\:\int\:\:\:\frac{{dx}}{{x}^{\mathrm{2}{n}} −\mathrm{1}}\:\:{with}\:{n}\:{integr}\:{natural}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 34222    Answers: 0   Comments: 4

let give the sequence of integrals J_n =∫_0 ^∞ x^n e^(−(x^2 /2)) dx 1) prove that J_n =(n−1)J_(n−2) ∀n≥2 2) calculate J_(2p) and J_(2p+1) by using factoriels. 3) prove that ∀n≥1 J_n ^2 ≤J_(n−1) . J_(n+1) . 4)prove that ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p+1))) ≤J_0 ≤ ((2^(2p) (p!)^2 )/((2p)!)) (1/(√(2p))) 5) find a equivalent of ((2^(2p) (p!)^2 )/((2p)!)) (p→+∞)

$${let}\:{give}\:{the}\:{sequence}\:{of}\:{integrals} \\ $$$${J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:{x}^{{n}} \:\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} {dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{J}_{{n}} =\left({n}−\mathrm{1}\right){J}_{{n}−\mathrm{2}} \:\:\:\forall{n}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{J}_{\mathrm{2}{p}} \:{and}\:{J}_{\mathrm{2}{p}+\mathrm{1}} \:{by}\:{using}\:{factoriels}. \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:\forall{n}\geqslant\mathrm{1}\:\:\:{J}_{{n}} ^{\mathrm{2}} \:\:\leqslant{J}_{{n}−\mathrm{1}} \:.\:{J}_{{n}+\mathrm{1}} . \\ $$$$\left.\mathrm{4}\right){prove}\:{that}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}+\mathrm{1}}}\:\leqslant{J}_{\mathrm{0}} \:\leqslant\:\frac{\mathrm{2}^{\mathrm{2}{p}} \:\left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\frac{\mathrm{1}}{\sqrt{\mathrm{2}{p}}} \\ $$$$\left.\mathrm{5}\right)\:{find}\:{a}\:{equivalent}\:{of}\:\:\frac{\mathrm{2}^{\mathrm{2}{p}} \left({p}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{p}\right)!}\:\:\left({p}\rightarrow+\infty\right) \\ $$

Question Number 34221    Answers: 1   Comments: 1

study the convergence of ∫_0 ^1 ((√(1−x))/x) dx .

$${study}\:{the}\:{convergence}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\sqrt{\mathrm{1}−{x}}}{{x}}\:{dx}\:. \\ $$

Question Number 34220    Answers: 0   Comments: 0

calculate I = ∫_0 ^(π/4) cosx ln(tanx)dx .

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cosx}\:{ln}\left({tanx}\right){dx}\:. \\ $$

Question Number 34219    Answers: 0   Comments: 1

calculate ∫_0 ^(π/4) (dx/(cos^3 x +sin^3 x))

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dx}}{{cos}^{\mathrm{3}} {x}\:+{sin}^{\mathrm{3}} {x}} \\ $$

Question Number 34218    Answers: 0   Comments: 0

find ∫(√(tanx))dx 2) calculate ∫_0 ^(π/6) (√(tanx)) dx

$${find}\:\int\sqrt{{tanx}}{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \sqrt{{tanx}}\:{dx} \\ $$

Question Number 34216    Answers: 0   Comments: 0

let give I =∫_0 ^1 ((ln(x+1))/x)dx and J = ∫_0 ^1 ((ln(1−x))/x)dx 1) prove the existence of I and J 2) calculate I +J and 2I +J 3) find I and J .

$${let}\:{give}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{I}\:{and}\:{J} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:+{J}\:{and}\:\mathrm{2}{I}\:+{J} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}\:{and}\:{J}\:. \\ $$

Question Number 34129    Answers: 2   Comments: 2

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

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

Question Number 34126    Answers: 0   Comments: 0

let give n natural integr not o calculate A_n = ∫_0 ^∞ (dx/(Π_(k=1) ^n (x^2 +k))) .

$${let}\:{give}\:{n}\:{natural}\:{integr}\:{not}\:{o} \\ $$$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}^{\mathrm{2}} \:+{k}\right)}\:. \\ $$

Question Number 34110    Answers: 0   Comments: 2

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