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IntegrationQuestion and Answers: Page 194

Question Number 85360    Answers: 1   Comments: 0

∫(e^((1−x)×e^x ) ×e^(∫xe^x dx) )dx

$$\int\left(\mathrm{e}^{\left(\mathrm{1}−\mathrm{x}\right)×\mathrm{e}^{\mathrm{x}} } ×\mathrm{e}^{\int\mathrm{xe}^{\mathrm{x}} \mathrm{dx}} \right)\mathrm{dx} \\ $$

Question Number 85355    Answers: 1   Comments: 4

Question Number 85374    Answers: 0   Comments: 2

∫_0 ^1 ((x^7 +x^3 +1)/(x^4 +1)) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{7}} +{x}^{\mathrm{3}} +\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}\:{dx}\: \\ $$

Question Number 85323    Answers: 1   Comments: 0

∫((z−3)/(5z−10))dz

$$\int\frac{\mathrm{z}−\mathrm{3}}{\mathrm{5z}−\mathrm{10}}\mathrm{dz} \\ $$

Question Number 85321    Answers: 0   Comments: 0

Question Number 85319    Answers: 1   Comments: 0

∫((2+u)/(−u^2 −u))du

$$\int\frac{\mathrm{2}+\mathrm{u}}{−\mathrm{u}^{\mathrm{2}} −\mathrm{u}}\mathrm{du} \\ $$

Question Number 85260    Answers: 1   Comments: 1

Evaluate using cauchy′s integral ∫_c (e^(iπ) /((z^2 +4)^2 (z+1)^2 ))dz where c is a circle with ∣z−i∣=3.5 help please

$$\mathrm{Evaluate}\:\mathrm{using}\:\mathrm{cauchy}'\mathrm{s}\:\mathrm{integral}\: \\ $$$$\:\:\:\:\:\:\:\:\:\int_{\mathrm{c}} \:\frac{\mathrm{e}^{\mathrm{i}\pi} }{\left(\mathrm{z}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} \left(\mathrm{z}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dz} \\ $$$$\mathrm{where}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mid\mathrm{z}−\mathrm{i}\mid=\mathrm{3}.\mathrm{5} \\ $$$$ \\ $$$$\mathrm{help}\:\mathrm{please} \\ $$

Question Number 85256    Answers: 0   Comments: 3

∫ _0 ^( 1) ((sin (ln x))/(ln (x))) dx

$$\int\underset{\mathrm{0}} {\overset{\:\mathrm{1}} {\:}}\:\frac{\mathrm{sin}\:\left(\mathrm{ln}\:\mathrm{x}\right)}{\mathrm{ln}\:\left(\mathrm{x}\right)}\:\mathrm{dx}\: \\ $$

Question Number 85254    Answers: 0   Comments: 0

Question Number 85236    Answers: 1   Comments: 1

find f(x) if f ′(x) + f(x^2 ) = 2x+1

$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{if}\: \\ $$$$\mathrm{f}\:'\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)\:=\:\mathrm{2x}+\mathrm{1} \\ $$

Question Number 85191    Answers: 2   Comments: 0

∫((z+2)/z)

$$\int\frac{\mathrm{z}+\mathrm{2}}{\mathrm{z}} \\ $$

Question Number 85167    Answers: 0   Comments: 0

let ϕ(x)=Γ(x).Γ(1−x) find ∫_(1/3) ^(1/2) ln(ϕ(x))dx

$${let}\:\varphi\left({x}\right)=\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:\:{find}\:\int_{\frac{\mathrm{1}}{\mathrm{3}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {ln}\left(\varphi\left({x}\right)\right){dx} \\ $$

Question Number 85166    Answers: 1   Comments: 3

find ∫ (x^2 −1)(√(x^2 +1))dx

$${find}\:\int\:\:\left({x}^{\mathrm{2}} −\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$

Question Number 85162    Answers: 0   Comments: 1

1)find ∫ ln((√x)+(√(x+1)))dx 2) calculate ∫_0 ^1 ln((√x)+(√(x+1)))dx

$$\left.\mathrm{1}\right){find}\:\int\:{ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\sqrt{{x}}+\sqrt{{x}+\mathrm{1}}\right){dx} \\ $$

Question Number 85160    Answers: 1   Comments: 2

1) find f(a) =∫_0 ^∞ (dx/(x^4 +a)) with a>0 2) find g(a)=∫_0 ^∞ (dx/((x^4 +a)^2 )) 3) find value of integrals ∫_0 ^∞ (dx/(x^4 +1)) ,∫_0 ^∞ (dx/(2x^4 +8)) ∫_0 ^∞ (dx/((x^4 +1)^2 )) and ∫_0 ^∞ (dx/((2x^4 +8)^2 ))

$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+{a}}\:{with}\:{a}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+{a}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{find}\:{value}\:{of}\:{integrals}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{{x}^{\mathrm{4}} \:+\mathrm{1}}\:,\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\mathrm{2}{x}^{\mathrm{4}} \:+\mathrm{8}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\frac{{dx}}{\left({x}^{\mathrm{4}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dx}}{\left(\mathrm{2}{x}^{\mathrm{4}} +\mathrm{8}\right)^{\mathrm{2}} } \\ $$

Question Number 85158    Answers: 0   Comments: 0

calculate U_n = ∫_(−(1/n)) ^(1/n) x^2 (√((1−x)/(1+x)))dx (n integr and n≥2) 2) find nature of Σ U_n

$${calculate}\:{U}_{{n}} =\:\int_{−\frac{\mathrm{1}}{{n}}} ^{\frac{\mathrm{1}}{{n}}} \:{x}^{\mathrm{2}} \sqrt{\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}}{dx}\:\:\:\left({n}\:{integr}\:{and}\:{n}\geqslant\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

Question Number 85148    Answers: 0   Comments: 0

∫_0 ^1 ((1+x^4 )/(1+x^3 +x^7 )) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}+{x}^{\mathrm{4}} }{\mathrm{1}+{x}^{\mathrm{3}} +{x}^{\mathrm{7}} }\:{dx} \\ $$

Question Number 85097    Answers: 1   Comments: 0

∫_(−π) ^π x^(2020) (sin x+cos x) dx = 8 find ∫_(−π) ^π x^(2020) cos x dx = ?

$$\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{8} \\ $$$$\mathrm{find}\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:=\:? \\ $$

Question Number 85059    Answers: 1   Comments: 0

Question Number 85009    Answers: 1   Comments: 1

calculate ∫_0 ^∞ (x^n /(sh(x)))dx with n integr natural

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{{n}} }{{sh}\left({x}\right)}{dx}\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 84958    Answers: 0   Comments: 0

Question Number 84957    Answers: 0   Comments: 1

∫ (2−x^2 )^3 dx =

$$\int\:\left(\mathrm{2}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{3}} \:\mathrm{dx}\:=\: \\ $$

Question Number 84956    Answers: 1   Comments: 3

show that ∫_0 ^(+∞) (1/(x^4 +2x^2 cos(((2π)/5))+1)) dx=(π/(2φ))

$${show}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{2}} {cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)+\mathrm{1}}\:{dx}=\frac{\pi}{\mathrm{2}\phi} \\ $$

Question Number 84942    Answers: 1   Comments: 0

∫_0 ^x sinh(x−t) cosh(t) dt

$$\int_{\mathrm{0}} ^{{x}} {sinh}\left({x}−{t}\right)\:{cosh}\left({t}\right)\:{dt} \\ $$

Question Number 84894    Answers: 0   Comments: 1

Question Number 84879    Answers: 2   Comments: 1

e^(∫((2dx)/(xlnx)))

$$\mathrm{e}^{\int\frac{\mathrm{2dx}}{\mathrm{xlnx}}} \\ $$

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