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IntegrationQuestion and Answers: Page 114

Question Number 129991    Answers: 0   Comments: 0

∫ (e^x )(((x^6 +x^5 +5x^4 )/((1+x)^6 )))dx = ...

$$\:\int\:\left(\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{x}}} \right)\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{6}} +\boldsymbol{\mathrm{x}}^{\mathrm{5}} +\mathrm{5}\boldsymbol{\mathrm{x}}^{\mathrm{4}} }{\left(\mathrm{1}+\boldsymbol{\mathrm{x}}\right)^{\mathrm{6}} }\right)\boldsymbol{\mathrm{dx}}\:=\:... \\ $$

Question Number 129989    Answers: 1   Comments: 0

Find the area bounded xy^2 = 4a^2 (2a−x) and its asymptotes.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded}\:\mathrm{xy}^{\mathrm{2}} \:=\:\mathrm{4a}^{\mathrm{2}} \left(\mathrm{2a}−\mathrm{x}\right) \\ $$$$\mathrm{and}\:\mathrm{its}\:\mathrm{asymptotes}. \\ $$

Question Number 129978    Answers: 1   Comments: 0

is this true for n∈N^★ ? someone please prove or falsify! ∫_0 ^∞ e^(−x^(2n) ) dx=Γ ((2n+1)/(2n))

$$\mathrm{is}\:\mathrm{this}\:\mathrm{true}\:\mathrm{for}\:{n}\in\mathbb{N}^{\bigstar} ?\:\mathrm{someone}\:\mathrm{please}\:\mathrm{prove} \\ $$$$\mathrm{or}\:\mathrm{falsify}! \\ $$$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\mathrm{e}^{−{x}^{\mathrm{2}{n}} } {dx}=\Gamma\:\frac{\mathrm{2}{n}+\mathrm{1}}{\mathrm{2}{n}} \\ $$

Question Number 129975    Answers: 2   Comments: 0

Question Number 129972    Answers: 1   Comments: 0

....nice calculus... evaluation: Ω=∫_0 ^( ∞) t^2 e^(−t) ln(t)dt=?? solution: f(s)=∫_0 ^( ∞) t^(2+s) e^(−t) dt Ω=f ′(0)=... f(s)=Γ(3+s) f ′(s)=Γ′(3+s)=ψ(3+s)Γ(3+s) f ′(0)=ψ(3)Γ(3)=2((3/2) −γ) =3−2γ ∴ Ω=∫_0 ^( ∞) t^2 e^(−t) ln(t)=3−2γ ...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:{calculus}... \\ $$$$\:\:{evaluation}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {t}^{\mathrm{2}} {e}^{−{t}} {ln}\left({t}\right){dt}=?? \\ $$$$\:\:{solution}: \\ $$$$\:\:\:{f}\left({s}\right)=\int_{\mathrm{0}} ^{\:\infty} {t}^{\mathrm{2}+{s}} {e}^{−{t}} {dt} \\ $$$$\:\:\:\Omega={f}\:'\left(\mathrm{0}\right)=... \\ $$$$\:\:\:{f}\left({s}\right)=\Gamma\left(\mathrm{3}+{s}\right) \\ $$$$\:\:\:\:{f}\:'\left({s}\right)=\Gamma'\left(\mathrm{3}+{s}\right)=\psi\left(\mathrm{3}+{s}\right)\Gamma\left(\mathrm{3}+{s}\right) \\ $$$${f}\:'\left(\mathrm{0}\right)=\psi\left(\mathrm{3}\right)\Gamma\left(\mathrm{3}\right)=\mathrm{2}\left(\frac{\mathrm{3}}{\mathrm{2}}\:−\gamma\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{3}−\mathrm{2}\gamma \\ $$$$\:\:\:\therefore\:\Omega=\int_{\mathrm{0}} ^{\:\infty} {t}^{\mathrm{2}} {e}^{−{t}} {ln}\left({t}\right)=\mathrm{3}−\mathrm{2}\gamma\:... \\ $$$$\:\:\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\:\: \\ $$

Question Number 129946    Answers: 1   Comments: 0

∫ ((tan ϕ+3)/(sin ϕ)) dϕ ?

$$\:\int\:\frac{\mathrm{tan}\:\varphi+\mathrm{3}}{\mathrm{sin}\:\varphi}\:\mathrm{d}\varphi\:? \\ $$

Question Number 129926    Answers: 1   Comments: 0

...nice calculus... evaluate: Σ_(k=0) ^∞ Σ_(m=0) ^∞ Σ_(n=0) ^∞ ((1/((k+m+n)!)))=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:\:{evaluate}: \\ $$$$\:\:\:\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\underset{{m}=\mathrm{0}} {\overset{\infty} {\sum}}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\left({k}+{m}+{n}\right)!}\right)=? \\ $$$$ \\ $$

Question Number 129921    Answers: 3   Comments: 0

∫(1/((1−u^2 )^2 u^5 ))du=?? please...

$$\int\frac{\mathrm{1}}{\left(\mathrm{1}−\mathrm{u}^{\mathrm{2}} \right)^{\mathrm{2}} \mathrm{u}^{\mathrm{5}} }\mathrm{du}=?? \\ $$$$\mathrm{please}... \\ $$

Question Number 129907    Answers: 3   Comments: 0

∫(dx/(sin^3 xcos^5 x))=???????

$$\int\frac{\boldsymbol{{dx}}}{\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{xcos}}^{\mathrm{5}} \boldsymbol{{x}}}=??????? \\ $$

Question Number 129936    Answers: 2   Comments: 0

Nice integral ∫_0 ^( ∞) sin (x) ln (x) e^(−x) dx

$$\:\mathrm{Nice}\:\mathrm{integral}\: \\ $$$$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{sin}\:\left({x}\right)\:\mathrm{ln}\:\left({x}\right)\:\mathrm{e}^{−{x}} \:{dx}\: \\ $$

Question Number 129868    Answers: 1   Comments: 0

find lim_(n→+∞) (1/n)Σ_(k=0) ^(n−1) (k/( (√(4n^2 −k^2 ))))

$${find}\:{lim}_{{n}\rightarrow+\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\frac{{k}}{\:\sqrt{\mathrm{4}{n}^{\mathrm{2}} −{k}^{\mathrm{2}} }} \\ $$

Question Number 129867    Answers: 1   Comments: 0

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

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

Question Number 129859    Answers: 3   Comments: 0

L = ∫_(−1) ^( 0) (√((1+y)/(1−y))) dy

$$\:\mathrm{L}\:=\:\int_{−\mathrm{1}} ^{\:\mathrm{0}} \sqrt{\frac{\mathrm{1}+\mathrm{y}}{\mathrm{1}−\mathrm{y}}}\:\mathrm{dy}\: \\ $$

Question Number 129855    Answers: 2   Comments: 0

ϑ = ∫ (dx/((1+(√x) )^3 ))

$$\:\vartheta\:=\:\int\:\frac{{dx}}{\left(\mathrm{1}+\sqrt{{x}}\:\right)^{\mathrm{3}} } \\ $$

Question Number 129839    Answers: 7   Comments: 0

Question Number 129816    Answers: 1   Comments: 0

Question Number 129788    Answers: 1   Comments: 0

∫ (1+3x^3 )e^x^3 dx

$$\:\int\:\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{3}} \right){e}^{{x}^{\mathrm{3}} } \:{dx}\: \\ $$

Question Number 129787    Answers: 1   Comments: 0

∫ x^7 (√(1+x^4 )) dx

$$\:\int\:{x}^{\mathrm{7}} \:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx}\: \\ $$

Question Number 129764    Answers: 2   Comments: 0

prove that ∫_(−∞) ^(+∞) x^2 e^(−x^2 ) cos(x^2 )sin(x^2 ) dx =(((√π)sin[(((√3)tan^(−1) (2))/2)])/(4 ((125))^(1/4) ))

$${prove}\:{that} \\ $$$$\int_{−\infty} ^{+\infty} {x}^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{2}} } \:{cos}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{\mathrm{2}} \right)\:{dx} \\ $$$$=\frac{\sqrt{\pi}{sin}\left[\frac{\sqrt{\mathrm{3}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)}{\mathrm{2}}\right]}{\mathrm{4}\:\sqrt[{\mathrm{4}}]{\mathrm{125}}} \\ $$

Question Number 129746    Answers: 2   Comments: 0

N = ∫ ((3+2cos x)/(2+3cos x)) dx

$$\:\mathrm{N}\:=\:\int\:\frac{\mathrm{3}+\mathrm{2cos}\:\mathrm{x}}{\mathrm{2}+\mathrm{3cos}\:\mathrm{x}}\:\mathrm{dx}\: \\ $$

Question Number 129763    Answers: 2   Comments: 0

prove ∫_(−∞) ^(+∞) (1/(1+e^x^2 ))dx=(√π) (1−(√2) )ξ((1/2))

$${prove} \\ $$$$\int_{−\infty} ^{+\infty} \frac{\mathrm{1}}{\mathrm{1}+{e}^{{x}^{\mathrm{2}} } }{dx}=\sqrt{\pi}\:\left(\mathrm{1}−\sqrt{\mathrm{2}}\:\right)\xi\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 129696    Answers: 1   Comments: 5

nice old question by sir m?th+et?s ∫_0 ^∞ cos((x^3 /3)+tx)dx

$${nice}\:{old}\:{question}\:{by}\:{sir}\:{m}?{th}+{et}?{s}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} {cos}\left(\frac{{x}^{\mathrm{3}} }{\mathrm{3}}+{tx}\right){dx} \\ $$$$ \\ $$

Question Number 129794    Answers: 3   Comments: 0

∫ (x^2 −1)(x+1)^(−2/3) dx ?

$$\:\:\int\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)^{−\mathrm{2}/\mathrm{3}} \:\mathrm{dx}\:? \\ $$

Question Number 129688    Answers: 1   Comments: 0

complex analysis ∮_C ((ϱ^(2z) +sinz^2 )/((z−2)(z−3)))dz C:∣Z∣=5

$${complex}\:{analysis} \\ $$$$\oint_{{C}} \frac{\varrho^{\mathrm{2}{z}} +{sinz}^{\mathrm{2}} }{\left({z}−\mathrm{2}\right)\left({z}−\mathrm{3}\right)}{dz}\:\:\:{C}:\mid{Z}\mid=\mathrm{5} \\ $$

Question Number 129684    Answers: 3   Comments: 0

... advanced calculus... prove that: ∫_0 ^( π) cos(tan(x)−cot(x))dx=(π/e^2 )

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:{calculus}... \\ $$$$\:\:{prove}\:\:{that}: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\pi} {cos}\left({tan}\left({x}\right)−{cot}\left({x}\right)\right){dx}=\frac{\pi}{{e}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 129651    Answers: 0   Comments: 0

... nice calculus... evaluate :: Σ_(n=1) ^∞ (((−1)^n )/(n^2 +1)) (((sin(n))/n))^2 =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:\:{calculus}... \\ $$$$\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}} +\mathrm{1}}\:\left(\frac{{sin}\left({n}\right)}{{n}}\right)^{\mathrm{2}} =? \\ $$$$ \\ $$

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