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IntegrationQuestion and Answers: Page 297

Question Number 35218 Answers: 0 Comments: 0

prove that ∫_0 ^∞ (t^(a−1) /(1+t))dt =(π/(sin(πa))) that we know 0<a<1 .

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

Question Number 35217 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((x sin(2x))/(x^2 +4))dx

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

Question Number 35215 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cosx +cos(2x))/(x^2 +9))dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cosx}\:+{cos}\left(\mathrm{2}{x}\right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$

Question Number 35214 Answers: 0 Comments: 0

let a>0 b ∈C and Re(b)>0 cslculate ∫_(−∞) ^(+∞) (e^(iax) /(x−ib))dx and ∫_(−∞) ^(+∞) (e^(iax) /(x+ib))dx

$${let}\:{a}>\mathrm{0}\:\:{b}\:\in{C}\:{and}\:{Re}\left({b}\right)>\mathrm{0} \\ $$$${cslculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{e}^{{iax}} }{{x}−{ib}}{dx}\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{{iax}} }{{x}+{ib}}{dx} \\ $$

Question Number 35213 Answers: 0 Comments: 0

find the values of ∫_0 ^∞ cos(λx^2 )dx and ∫_0 ^∞ sin(λx^2 )dx with λ>0 . 2) find the values of ∫_0 ^∞ cos(x^2 )dx and ∫_0 ^∞ sin(x^2 )dx( integrals of fresnel)

$${find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left(\lambda{x}^{\mathrm{2}} \right){dx}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{sin}\left(\lambda{x}^{\mathrm{2}} \right){dx}\:{with}\:\lambda>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\left(\:{integrals}\:{of}\:{fresnel}\right) \\ $$

Question Number 35212 Answers: 0 Comments: 0

prove by using series only that ∫_0 ^∞ cos(x^2 )dx= ∫_0 ^∞ sin(x^2 )dx.

$${prove}\:{by}\:{using}\:{series}\:{only}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}=\:\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{\mathrm{2}} \right){dx}. \\ $$

Question Number 35202 Answers: 1 Comments: 0

∫^ e^(3x) (√(1−e^(2x) ))dx plzz help

$$\:\int^{} \:\boldsymbol{{e}}^{\mathrm{3}\boldsymbol{{x}}} \sqrt{\mathrm{1}−\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}} }\boldsymbol{{dx}} \\ $$$$\boldsymbol{{plzz}}\:\boldsymbol{{help}} \\ $$

Question Number 35186 Answers: 0 Comments: 0

find ∫_0 ^π ((x dx)/(3 +cosx))

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{x}\:{dx}}{\mathrm{3}\:+{cosx}} \\ $$

Question Number 35139 Answers: 1 Comments: 2

Question Number 35117 Answers: 1 Comments: 0

∫((2x+3)/(x^4 −3x−2))dx

$$\int\frac{\mathrm{2}{x}+\mathrm{3}}{{x}^{\mathrm{4}} −\mathrm{3}{x}−\mathrm{2}}{dx} \\ $$

Question Number 35101 Answers: 2 Comments: 1

Find volume enclosed by (x^2 /a^2 )+(y^2 /b^2 )+(z^2 /c^2 )=1 .

$${Find}\:{volume}\:{enclosed}\:{by} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }+\frac{{z}^{\mathrm{2}} }{{c}^{\mathrm{2}} }=\mathrm{1}\:\:. \\ $$

Question Number 35062 Answers: 0 Comments: 0

calculate A_n = ∫_0 ^∞ (dt/((t^4 +1)^n )) with n integr natural .

$${calculate}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+\mathrm{1}\right)^{{n}} } \\ $$$${with}\:{n}\:{integr}\:{natural}\:. \\ $$

Question Number 35061 Answers: 2 Comments: 1

find ∫_0 ^∞ ((x^2 +3)/((x^4 +1)^2 ))dx

$${find}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{x}^{\mathrm{2}} \:+\mathrm{3}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 35060 Answers: 1 Comments: 1

calculate ∫_0 ^(π/4) sinx ln(cosx)dx

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

Question Number 35059 Answers: 2 Comments: 2

find ∫_0 ^π (dx/(cosx +sinx))

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\frac{{dx}}{{cosx}\:+{sinx}} \\ $$

Question Number 35058 Answers: 1 Comments: 1

find ∫_0 ^(π/4) (dt/((1+cos^2 t)^3 ))

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{cos}^{\mathrm{2}} {t}\right)^{\mathrm{3}} } \\ $$

Question Number 35055 Answers: 1 Comments: 2

calculate ∫_(−∞) ^(+∞) (dx/((1+x+x^2 )^3 ))

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

Question Number 35054 Answers: 0 Comments: 1

find ∫_0 ^(π/4) ((xdx)/(2 +cosx))

$${find}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{xdx}}{\mathrm{2}\:+{cosx}} \\ $$

Question Number 35053 Answers: 0 Comments: 0

let v(x)=ln(1+x+x^2 ) developp f at integr serie.

$${let}\:{v}\left({x}\right)={ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right) \\ $$$${developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 35049 Answers: 1 Comments: 1

let A_n = ∫_(1/n) ^n (1+(1/x^2 ))arctanx dx 1) calculate A_n 2) find lim_(n→+∞) A_n

$${let}\:{A}_{{n}} \:=\:\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctanx}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$

Question Number 35048 Answers: 0 Comments: 0

find ∫ (dx/(cos(sinx)))

$${find}\:\int\:\:\:\:\:\frac{{dx}}{{cos}\left({sinx}\right)} \\ $$

Question Number 35046 Answers: 0 Comments: 0

find F(x)= ∫_0 ^π ln( 1+x sin^2 t)dt with ∣x∣<1 2) calculate ∫_0 ^π ln(1+(1/2)sin^2 t)dt

$${find}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:{ln}\left(\:\mathrm{1}+{x}\:{sin}^{\mathrm{2}} {t}\right){dt}\:{with}\:\mid{x}\mid<\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{sin}^{\mathrm{2}} {t}\right){dt} \\ $$

Question Number 35045 Answers: 0 Comments: 0

find f(x)=∫_0 ^∞ ((arctan(xt))/(1+t^2 ))dt .

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

Question Number 35044 Answers: 1 Comments: 1

1)find ∫ (√(1+t^2 )) dt 2) calculate ∫_1 ^(√3) (√(1+t^2 )) dt

$$\left.\mathrm{1}\right){find}\:\int\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:\:{dt} \\ $$

Question Number 35043 Answers: 1 Comments: 0

let t>0 and F(t) =∫_0 ^∞ ((sin(x^2 ) e^(−tx^2 ) )/x^2 )dx calculate (dF/dt)(t).

$${let}\:{t}>\mathrm{0}\:{and}\:{F}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)\:{e}^{−{tx}^{\mathrm{2}} } }{{x}^{\mathrm{2}} }{dx} \\ $$$${calculate}\:\frac{{dF}}{{dt}}\left({t}\right). \\ $$

Question Number 35018 Answers: 1 Comments: 0

∫∫∫((dxdydz)/((x+y+z+1)^3 )) bounded by the coordinate planes and the plane x+y+z=1 .

$$\int\int\int\frac{{dxdydz}}{\left({x}+{y}+{z}+\mathrm{1}\right)^{\mathrm{3}} }\:\:\:{bounded}\:{by}\:{the} \\ $$$${coordinate}\:{planes}\:{and}\:{the}\:{plane} \\ $$$${x}+{y}+{z}=\mathrm{1}\:. \\ $$

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