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IntegrationQuestion and Answers: Page 104

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 =?

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

Question Number 129645 Answers: 1 Comments: 0

ϝ = ∫ (dx/(1+x^(12) )) .

$$\:\:\:\:\digamma\:=\:\int\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{12}} }\:. \\ $$

Question Number 129646 Answers: 3 Comments: 0

ϝ = ∫ (dx/(x^(2n+1) (x^n −1)))

$$\:\:\digamma\:=\:\int\:\frac{{dx}}{{x}^{\mathrm{2}{n}+\mathrm{1}} \left({x}^{{n}} −\mathrm{1}\right)} \\ $$

Question Number 129635 Answers: 3 Comments: 1

∫(dx/((x^2 +1)^2 ))=? Σ_(k=1) ^n (1/(k(k+1)(2k+1)))=?

$$\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\int\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }=? \\ $$$$\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\frac{\mathrm{1}}{\boldsymbol{{k}}\left(\boldsymbol{{k}}+\mathrm{1}\right)\left(\mathrm{2}\boldsymbol{{k}}+\mathrm{1}\right)}=? \\ $$

Question Number 129576 Answers: 1 Comments: 0

please, how to show that f : [0 , a] × R_+ → R (x, y) e^(−xy) sin x is integrable ???

$$\boldsymbol{\mathrm{please}},\:\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{f}}\::\:\left[\mathrm{0}\:,\:\boldsymbol{{a}}\right]\:×\:\mathbb{R}_{+} \:\rightarrow\:\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{{x}},\:\boldsymbol{{y}}\right)\:\:\:\:\:\:\: \:\:\boldsymbol{{e}}^{−\boldsymbol{{xy}}} \:\boldsymbol{{sin}}\:\boldsymbol{{x}}\: \\ $$$$\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{integrable}}\:???\: \\ $$

Question Number 129564 Answers: 2 Comments: 0

V = ∫ ((sin x)/(sin (x+θ))) dx

$$\:\mathcal{V}\:=\:\int\:\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\left(\mathrm{x}+\theta\right)}\:\mathrm{dx}\: \\ $$

Question Number 129562 Answers: 1 Comments: 0

calculate ∫_0 ^∞ (dx/((x^6 +1)^2 ))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{6}} +\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 129558 Answers: 1 Comments: 0

...modern ∗∗∗∗∗∗∗∗∗∗ algebra ... ::: if ′′ G ′′ be a finite group and O (G)=pq , where ′′ p , q ′′ are two prime numbers (p > q ) then prove that: G has at most one subgroup of order ′′ p ′′ . written and compiled by ...♣m.n.july.1970♣....

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{modern}\:\ast\ast\ast\ast\ast\ast\ast\ast\ast\ast\:{algebra}\:...\: \\ $$$$\:\:\:\:\:\:\:\::::\:\:{if}\:\:''\:{G}\:''\:{be}\:{a}\:{finite}\:{group}\:{and} \\ $$$$\:\:{O}\:\left({G}\right)={pq}\:\:,\:\:{where}\:''\:{p}\:,\:{q}\:''\:{are}\:{two} \\ $$$$\:\:{prime}\:\:{numbers}\:\left({p}\:>\:{q}\:\right)\:{then}\:{prove}\:{that}: \\ $$$$\:\:{G}\:\:{has}\:\:{at}\:{most}\:{one}\:{subgroup}\:{of}\:{order}\:''\:{p}\:''\:. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{written}\:{and}\:{compiled}\:{by} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\clubsuit{m}.{n}.{july}.\mathrm{1970}\clubsuit.... \\ $$

Question Number 129594 Answers: 0 Comments: 1

Question Number 129519 Answers: 1 Comments: 0

∫ cos (y^3 )dy

$$\int\:{cos}\:\left({y}^{\mathrm{3}} \right){dy} \\ $$

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