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IntegrationQuestion and Answers: Page 292

Question Number 36186    Answers: 0   Comments: 1

find nature of ∫_1 ^(+∞) (√t) sin(t^2 )dt .

$${find}\:{nature}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \sqrt{{t}}\:{sin}\left({t}^{\mathrm{2}} \right){dt}\:. \\ $$

Question Number 36185    Answers: 0   Comments: 2

study the vonvergence of ∫_1 ^(+∞) ((e^(−(1/t)) −cos((1/t)))/t)dt

$${study}\:{the}\:{vonvergence}\:{of}\: \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{−\frac{\mathrm{1}}{{t}}} \:−{cos}\left(\frac{\mathrm{1}}{{t}}\right)}{{t}}{dt} \\ $$

Question Number 36184    Answers: 0   Comments: 1

study the convergence of ∫_1 ^(+∞) ((cos(t))/(√t))dt

$${study}\:{the}\:{convergence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{cos}\left({t}\right)}{\sqrt{{t}}}{dt} \\ $$

Question Number 36183    Answers: 0   Comments: 0

calculate ∫_1 ^(+∞) arctan((1/t))dt

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:{arctan}\left(\frac{\mathrm{1}}{{t}}\right){dt} \\ $$

Question Number 36182    Answers: 2   Comments: 1

calculate ∫_1 ^(+∞) (dt/(t(√(1+t^2 ))))

$${calculate}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dt}}{{t}\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }} \\ $$

Question Number 36181    Answers: 0   Comments: 1

let I(ξ) = ∫_ξ ^(1−ξ) (dt/(1−(t−ξ)^2 )) find lim_(ξ→0^+ ) I(ξ)

$${let}\:{I}\left(\xi\right)\:\:=\:\int_{\xi} ^{\mathrm{1}−\xi} \:\:\:\frac{{dt}}{\mathrm{1}−\left({t}−\xi\right)^{\mathrm{2}} } \\ $$$${find}\:{lim}_{\xi\rightarrow\mathrm{0}^{+} } \:\:\:{I}\left(\xi\right) \\ $$

Question Number 36180    Answers: 1   Comments: 1

calculate ∫_0 ^1 ((ln(t))/((1+t)^2 ))dt

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

Question Number 36167    Answers: 0   Comments: 2

let give I = ∫_0 ^∞ (dx/((x^2 +i)^2 )) 1) extract Re(I) and Im(I) 2) find the value of I 3) calculate Re(I) and Im(I) .

$${let}\:{give}\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+{i}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{Re}\left({I}\right)\:{and}\:{Im}\left({I}\right)\:. \\ $$

Question Number 36057    Answers: 2   Comments: 1

find the value of ∫_0 ^(π/4) ((cosx)/(sinx +tanx))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{cosx}}{{sinx}\:+{tanx}}{dx}\: \\ $$

Question Number 36056    Answers: 1   Comments: 2

calculate ∫_(−∞) ^(+∞) ((2x)/((x^2 +mx +1)^2 ))dx with ∣m∣<2

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{2}{x}}{\left({x}^{\mathrm{2}} \:+{mx}\:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:{with}\:\mid{m}\mid<\mathrm{2} \\ $$

Question Number 36031    Answers: 0   Comments: 0

Q. Evaluate: ∫_(∫xyzdxdydz) ^(∫zyxdzdydx) ∫_((d/dx)(x^(sin x) )) ^((d/dx)(x^(cos x) )) ∫_(lim_(x→0) ((−x^2 +2)/x)) ^(lim_(x→0) ((x^2 −2)/x)) ∫_0 ^∞ w^(1−x) x^(1−y) y^(1−z) z^(1−w) dwdxdydz

$$\mathrm{Q}.\:\mathrm{Evaluate}:\:\:\:\int_{\int\mathrm{xyzdxdydz}} ^{\int\mathrm{zyxdzdydx}} \int_{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{sin}\:\mathrm{x}} \right)} ^{\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{cos}\:\mathrm{x}} \right)} \int_{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{x}}} ^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2}} −\mathrm{2}}{\mathrm{x}}} \int_{\mathrm{0}} ^{\infty} \mathrm{w}^{\mathrm{1}−\mathrm{x}} \mathrm{x}^{\mathrm{1}−\mathrm{y}} \mathrm{y}^{\mathrm{1}−\mathrm{z}} \mathrm{z}^{\mathrm{1}−\mathrm{w}} \mathrm{dwdxdydz} \\ $$

Question Number 36009    Answers: 0   Comments: 1

calculate ∫_(−∞) ^(+∞) ((xdx)/((2x+1+i)^3 )) with i^2 =−1 .

$${calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\:\:\frac{{xdx}}{\left(\mathrm{2}{x}+\mathrm{1}+{i}\right)^{\mathrm{3}} }\:\:{with}\:{i}^{\mathrm{2}} \:=−\mathrm{1}\:. \\ $$

Question Number 35990    Answers: 0   Comments: 2

calculate ∫_2 ^5 ((xdx)/(2x+1 +(√(x−1))))

$${calculate}\:\int_{\mathrm{2}} ^{\mathrm{5}} \:\:\:\:\frac{{xdx}}{\mathrm{2}{x}+\mathrm{1}\:+\sqrt{{x}−\mathrm{1}}} \\ $$

Question Number 35983    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((2(√t) +1)/(t^5 +3))dt .

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{\mathrm{2}\sqrt{{t}}\:+\mathrm{1}}{{t}^{\mathrm{5}} \:\:\:+\mathrm{3}}{dt}\:\:. \\ $$

Question Number 35982    Answers: 0   Comments: 0

let f(t) =∫_0 ^∞ e^(−arctsn( 1+tx^2 )) dx with t from R 1) calculate f^′ (t) 2) find a simple form of f(t) .

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−{arctsn}\left(\:\mathrm{1}+{tx}^{\mathrm{2}} \right)} {dx}\:\:{with}\:{t}\:{from}\:{R} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({t}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right)\:. \\ $$

Question Number 35949    Answers: 1   Comments: 1

∫_0 ^( α) ((tan θ)/(√(a^2 cos^2 θ−b^2 sin^2 θ))) dθ = ?

$$\int_{\mathrm{0}} ^{\:\:\alpha} \frac{\mathrm{tan}\:\theta}{\sqrt{{a}^{\mathrm{2}} \mathrm{cos}\:^{\mathrm{2}} \theta−{b}^{\mathrm{2}} \mathrm{sin}\:^{\mathrm{2}} \theta}}\:{d}\theta\:=\:? \\ $$

Question Number 35920    Answers: 1   Comments: 1

∫ ((e^(2x) +1)/(2e^x −1)) dx = ?

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

Question Number 35909    Answers: 1   Comments: 4

∫((7x−6)/((x^2 +25)(√((x−3)^2 +4)))) dx = ?

$$\int\frac{\mathrm{7}{x}−\mathrm{6}}{\left({x}^{\mathrm{2}} +\mathrm{25}\right)\sqrt{\left({x}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{4}}}\:{dx}\:=\:? \\ $$

Question Number 35832    Answers: 1   Comments: 3

find the value of f(λ) = ∫_(−a) ^a (dx/((λ +_ x^2 )^(3/2) )) λ∈R .

$${find}\:{the}\:{value}\:{of}\:\:{f}\left(\lambda\right)\:=\:\int_{−{a}} ^{{a}} \:\:\:\frac{{dx}}{\left(\lambda\:+_{} {x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\lambda\in{R}\:. \\ $$

Question Number 35821    Answers: 0   Comments: 4

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

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

Question Number 35732    Answers: 1   Comments: 1

Question Number 35729    Answers: 1   Comments: 1

find the value of I =∫_0 ^π (dx/(2−cosx))

$${find}\:{the}\:{value}\:{of}\:\:\:{I}\:=\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{2}−{cosx}} \\ $$

Question Number 35691    Answers: 0   Comments: 0

calculate lim_(a→0^+ ) ∫_(−a) ^a (√((1+x^2 )/(a^2 −x^2 ))) dx .

$${calculate}\:{lim}_{{a}\rightarrow\mathrm{0}^{+} \:\:\:\:} \:\:\:\int_{−{a}} ^{{a}} \:\:\sqrt{\frac{\mathrm{1}+{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} \:−{x}^{\mathrm{2}} }}\:\:{dx}\:. \\ $$

Question Number 35687    Answers: 1   Comments: 2

calculate f(a)=∫_0 ^π (dx/(1−a cosx)) a from R . 2) application calculate ∫_0 ^π (dx/(1−2cosx))

$${calculate}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}−{a}\:{cosx}}\:\:{a}\:{from}\:{R}\:. \\ $$$$\left.\mathrm{2}\right)\:{application}\:\:{calculate}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}−\mathrm{2}{cosx}} \\ $$

Question Number 35686    Answers: 1   Comments: 1

calculate ∫_(√3) ^(+∞) (dx/(x(√( 2+x^2 )))) .

$${calculate}\:\:\int_{\sqrt{\mathrm{3}}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{{x}\sqrt{\:\mathrm{2}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 35685    Answers: 1   Comments: 1

calculate ∫_0 ^(π/4) x artan(2x+1)dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:{x}\:{artan}\left(\mathrm{2}{x}+\mathrm{1}\right){dx} \\ $$

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