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Question Number 35471    Answers: 0   Comments: 0

∫_a ^b f(x)dx=area under the curve but say why what is the meaning of ∫ ←this sign

$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}={area}\:{under}\:{the}\:{curve}\:{but}\:{say}\:{why} \\ $$$${what}\:{is}\:{the}\:{meaning}\:{of}\:\int\:\leftarrow{this}\:{sign} \\ $$

Question Number 35456    Answers: 2   Comments: 0

∫(dx/(x(x^(2018) +1)))

$$\int\frac{{dx}}{{x}\left({x}^{\mathrm{2018}} +\mathrm{1}\right)} \\ $$

Question Number 35440    Answers: 1   Comments: 2

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

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

Question Number 35438    Answers: 0   Comments: 1

let m>0 and 0<a<b 1) calculate ∫_0 ^∞ ((cos(mx))/((x^2 +a^2 )(x^2 +b^2 )))dx 2)find the value of ∫_0 ^∞ ((cos(2x))/((x^2 +1)(x^2 +3)))dx

$${let}\:{m}>\mathrm{0}\:{and}\:\mathrm{0}<{a}<{b}\: \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{cos}\left({mx}\right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} \:+{b}^{\mathrm{2}} \right)}{dx} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} \:\:+\mathrm{3}\right)}{dx} \\ $$

Question Number 35429    Answers: 1   Comments: 1

find ∫_0 ^1 ((2x−1)/(√(x^2 +6))) dx

$${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{\mathrm{2}{x}−\mathrm{1}}{\sqrt{{x}^{\mathrm{2}} \:\:+\mathrm{6}}}\:{dx} \\ $$

Question Number 35428    Answers: 1   Comments: 1

find ∫ ((x+3)/(√(x^2 +x −1)))dx

$${find}\:\:\:\:\:\int\:\:\:\:\frac{{x}+\mathrm{3}}{\sqrt{{x}^{\mathrm{2}} \:+{x}\:−\mathrm{1}}}{dx} \\ $$

Question Number 35427    Answers: 1   Comments: 1

calculate ∫_2 ^5 (e^(√(x+1)) /(√(x+1)))dx

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

Question Number 35379    Answers: 2   Comments: 3

∫_0 ^( 1) t^2 (√(1+t^2 )) dt = ?

$$\int_{\mathrm{0}} ^{\:\:\mathrm{1}} {t}^{\mathrm{2}} \sqrt{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:=\:? \\ $$

Question Number 35325    Answers: 1   Comments: 0

Sketch the region enclosed by the curves of y=1/x and y=1/x^2 and find the area of the region. plzz help me

$${Sketch}\:{the}\:{region}\:{enclosed}\:{by}\:{the} \\ $$$${curves}\:{of}\:{y}=\mathrm{1}/{x}\:{and}\:{y}=\mathrm{1}/{x}^{\mathrm{2}} \:{and} \\ $$$${find}\:{the}\:{area}\:{of}\:{the}\:{region}. \\ $$$${plzz}\:{help}\:{me} \\ $$

Question Number 35294    Answers: 1   Comments: 1

Question Number 35290    Answers: 0   Comments: 1

∫((x+1)/x^3 )dx

$$\int\frac{{x}+\mathrm{1}}{{x}^{\mathrm{3}} }{dx} \\ $$

Question Number 35242    Answers: 1   Comments: 1

find ∫_0 ^π ((xdx)/(1+sinx))

$${find}\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{xdx}}{\mathrm{1}+{sinx}} \\ $$

Question Number 35241    Answers: 2   Comments: 6

calculate ∫_0 ^π ((x dx)/(3 +cosx)) .

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

Question Number 35238    Answers: 0   Comments: 1

study the convergence of ∫_1 ^(+∞) ((e^(−3x) −e^(−2x) )/x^2 )dx

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{e}^{−\mathrm{3}{x}} \:−{e}^{−\mathrm{2}{x}} }{{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 35237    Answers: 0   Comments: 1

study the convergence of ∫_0 ^∞ ((e^(−x) −e^(−x^2 ) )/x)dx .

$${study}\:{the}\:{convergence}\:{of} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}} \:−{e}^{−{x}^{\mathrm{2}} } }{{x}}{dx}\:. \\ $$

Question Number 35229    Answers: 1   Comments: 2

find the value of integral ∫_0 ^∞ e^(−(2+ia)^2 t^2 ) dt with a from R ∣a∣<1.

$${find}\:{the}\:{value}\:{of}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left(\mathrm{2}+{ia}\right)^{\mathrm{2}} {t}^{\mathrm{2}} } {dt}\:\:\:\:{with}\:{a}\:{from}\:{R}\:\:\:\:\mid{a}\mid<\mathrm{1}. \\ $$

Question Number 35228    Answers: 0   Comments: 2

find the value of integral ∫_0 ^∞ e^(−px) ((sin(qx))/(√x))dx with p>0 and q>0

$${find}\:{the}\:{value}\:{of}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{px}} \:\:\:\frac{{sin}\left({qx}\right)}{\sqrt{{x}}}{dx}\:\:{with}\:{p}>\mathrm{0}\:{and}\:{q}>\mathrm{0} \\ $$

Question Number 35226    Answers: 0   Comments: 4

1) calculate f(a) = ∫_0 ^π (dx/(a sin^2 x +cos^2 x)) with a>0 2) find the value of g(a) = ∫_0 ^π ((sin^2 x)/((a sin^2 x +cos^2 x)^2 ))dx

$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\:\:\frac{{dx}}{{a}\:{sin}^{\mathrm{2}} {x}\:\:+{cos}^{\mathrm{2}} {x}} \\ $$$${with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{sin}^{\mathrm{2}} {x}}{\left({a}\:{sin}^{\mathrm{2}} {x}\:+{cos}^{\mathrm{2}} {x}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 35225    Answers: 0   Comments: 4

1) find f(a) = ∫_0 ^(2π) (dt/(a cos^2 t + sin^2 t)) with a≠0 2) find g(a) = ∫_0 ^(2π) ((cos^2 t)/((a cos^2 t +sin^2 t)^2 ))dt

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\:\frac{{dt}}{{a}\:{cos}^{\mathrm{2}} {t}\:+\:{sin}^{\mathrm{2}} {t}}\:{with}\:{a}\neq\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}^{\mathrm{2}} {t}}{\left({a}\:{cos}^{\mathrm{2}} {t}\:+{sin}^{\mathrm{2}} {t}\right)^{\mathrm{2}} }{dt}\: \\ $$

Question Number 35224    Answers: 1   Comments: 0

calculate ∫_0 ^(2π) ((1+2cost)/(5+4cost))dt

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\:\frac{\mathrm{1}+\mathrm{2}{cost}}{\mathrm{5}+\mathrm{4}{cost}}{dt} \\ $$

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.

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

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