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IntegrationQuestion and Answers: Page 128

Question Number 123687    Answers: 1   Comments: 0

...nice calculus.. prove that:: lim_(x→0) (((2φ(x))/x^2 ) +(π^2 /(3x))) =^(???) ζ(3) where φ(x)=∫_0 ^( 1) (((t^x −1)(ln(1−t)))/(tln(t)))dt

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}.. \\ $$$$\:{prove}\:{that}::\:\:\:\:{lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{\mathrm{2}\phi\left({x}\right)}{{x}^{\mathrm{2}} }\:+\frac{\pi^{\mathrm{2}} }{\mathrm{3}{x}}\right)\:\overset{???} {=}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{where} \\ $$$$\:\:\:\:\:\phi\left({x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({t}^{{x}} −\mathrm{1}\right)\left({ln}\left(\mathrm{1}−{t}\right)\right)}{{tln}\left({t}\right)}{dt} \\ $$

Question Number 123639    Answers: 1   Comments: 0

∫_0 ^∞ (√(tan^(−1) (x))) dx ?

$$\:\:\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\sqrt{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}\:{dx}\:? \\ $$

Question Number 123552    Answers: 4   Comments: 0

∫_( 0) ^( (π/4)) tan^n (x)dx

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \mathrm{tan}^{\mathrm{n}} \left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 123550    Answers: 3   Comments: 0

....nice mathematics... evaluate :::: Ω=∫_0 ^( ∞) ((√(x+1)) −(√x) )^(10) dx=?

$$\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:\:{mathematics}... \\ $$$$\:\:\:\:\:\:{evaluate}\::::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \left(\sqrt{{x}+\mathrm{1}}\:−\sqrt{{x}}\:\right)^{\mathrm{10}} {dx}=? \\ $$

Question Number 123513    Answers: 3   Comments: 1

∫ ((sin x+2cos x)/(3sin x+4cos x)) dx?

$$\int\:\frac{\mathrm{sin}\:{x}+\mathrm{2cos}\:{x}}{\mathrm{3sin}\:{x}+\mathrm{4cos}\:{x}}\:{dx}? \\ $$

Question Number 123495    Answers: 1   Comments: 0

∫_0 ^∞ ((x−1)/((2−(√x))(1−x^3 ))) dx ?

$$\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{x}−\mathrm{1}}{\left(\mathrm{2}−\sqrt{{x}}\right)\left(\mathrm{1}−{x}^{\mathrm{3}} \right)}\:{dx}\:? \\ $$

Question Number 123494    Answers: 2   Comments: 0

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

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

Question Number 123462    Answers: 1   Comments: 0

...nice calculus... prove that :: Ω=∫_0 ^( 1) ((ln(1−x)ln(1−x^2 ))/x) =^(??) ((11ζ( 3 ))/8) .................

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:\:\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right){ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)}{{x}}\:\overset{??} {=}\frac{\mathrm{11}\zeta\left(\:\mathrm{3}\:\right)}{\mathrm{8}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:................. \\ $$

Question Number 123456    Answers: 0   Comments: 0

... advanced calculus... prove:: φ=Σ_(n=1) ^∞ ( (H_n /n^3 ) )=^(???) (π^4 /(72)) note: H_n =1+(1/2)+...+(1/n)

$$\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}:: \\ $$$$\:\:\:\:\:\:\:\:\:\phi=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\mathrm{H}_{{n}} }{{n}^{\mathrm{3}} }\:\right)\overset{???} {=}\frac{\pi^{\mathrm{4}} }{\mathrm{72}} \\ $$$$\:\:\:\:\:\:\:{note}:\:\mathrm{H}_{{n}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+...+\frac{\mathrm{1}}{{n}} \\ $$

Question Number 123452    Answers: 4   Comments: 1

∫(dx/( (x^2 +n)(√(x^2 +a))))

$$\int\frac{{dx}}{\:\left({x}^{\mathrm{2}} +{n}\right)\sqrt{{x}^{\mathrm{2}} +{a}}} \\ $$

Question Number 123454    Answers: 1   Comments: 0

...nice calculus... calculate ::: Ω =^(???) ∫_0 ^( ∞) (√x) Π_(n=1) ^∞ (cos((x/2^n )))dx

$$\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\:{calculate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\Omega\:\overset{???} {=}\int_{\mathrm{0}} ^{\:\infty} \sqrt{{x}}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\left({cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)\right){dx} \\ $$

Question Number 123526    Answers: 3   Comments: 2

∫_0 ^∞ ((x arctan x)/((1+x^2 )^2 )) dx ?

$$\:\:\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{{x}\:\mathrm{arctan}\:{x}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 123387    Answers: 0   Comments: 2

... nice calculus... prove that :: Ω=∫_0 ^( 1) (((ln(x))^2 li_3 (x))/(1−x)) dx =^(???) ζ^2 (3)−ζ(6) ✓

$$\:\:\:\:\:...\:\:{nice}\:{calculus}... \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{\mathrm{2}} {li}_{\mathrm{3}} \left({x}\right)}{\mathrm{1}−{x}}\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{???} {=}\zeta^{\mathrm{2}} \left(\mathrm{3}\right)−\zeta\left(\mathrm{6}\right)\:\checkmark \\ $$

Question Number 123386    Answers: 1   Comments: 0

Given f(x)=(∫_0 ^1 f(x)dx)x^2 +(∫_0 ^2 f(x)dx)x+(∫_0 ^3 f(x)dx)+1 then the value of f(4) = ...

$$\:{Given}\: \\ $$$${f}\left({x}\right)=\left(\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} +\left(\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}\right){x}+\left(\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}\right)+\mathrm{1} \\ $$$${then}\:{the}\:{value}\:{of}\:{f}\left(\mathrm{4}\right)\:=\:... \\ $$

Question Number 123361    Answers: 0   Comments: 1

please find the ∫(e^x /x)dx

$${please}\:{find}\:{the}\:\int\frac{{e}^{{x}} }{{x}}{dx} \\ $$

Question Number 123352    Answers: 0   Comments: 0

∫(e^x /x)dx

$$\int\frac{{e}^{{x}} }{{x}}{dx} \\ $$

Question Number 123331    Answers: 1   Comments: 0

∫ ((sinx)/x) dx

$$\int\:\frac{{sinx}}{{x}}\:{dx} \\ $$

Question Number 123261    Answers: 1   Comments: 0

... nice calculus... prove that:: Ω=∫_R e^(x−sinh^2 (x)) dx=(√π)

$$\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:{prove}\:\:{that}:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega=\int_{\mathbb{R}} {e}^{{x}−{sinh}^{\mathrm{2}} \left({x}\right)} {dx}=\sqrt{\pi} \\ $$

Question Number 123255    Answers: 1   Comments: 0

∗∗∗ nice calculus ∗∗∗ evaluate :: Φ=∫_0 ^(π/2) log^3 (tan(x))dx =?

$$\:\:\:\:\:\:\:\:\:\:\ast\ast\ast\:\:{nice}\:\:{calculus}\:\ast\ast\ast \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {log}^{\mathrm{3}} \left({tan}\left({x}\right)\right){dx}\:=? \\ $$

Question Number 123253    Answers: 0   Comments: 0

Question Number 123234    Answers: 3   Comments: 1

∫ (√(x^2 −4x+5)) dx

$$\:\:\int\:\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{5}}\:{dx}\: \\ $$

Question Number 123177    Answers: 0   Comments: 0

lebesgue measure on [0 1] is finite ? true or false give reason

$${lebesgue}\:{measure}\:{on}\:\left[\mathrm{0}\:\mathrm{1}\right]\:{is}\:{finite}\:?\:{true}\:{or}\:{false}\:{give}\:{reason} \\ $$

Question Number 123159    Answers: 3   Comments: 0

∫_( 0) ^( (π/2)) log^2 (tan(x))dx

$$\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{log}^{\mathrm{2}} \left(\mathrm{tan}\left(\mathrm{x}\right)\right)\mathrm{dx} \\ $$

Question Number 123154    Answers: 3   Comments: 0

Question Number 123060    Answers: 3   Comments: 0

.... nice calculus .... evaluate ::: Ω=^(???) ∫_(−∞) ^( ∞) (x^2 /((1+e^x )(1+e^(−x) )))dx

$$\:\:\:\:\:\:\:\:\:\:....\:\:\:{nice}\:\:{calculus}\:.... \\ $$$$\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega\overset{???} {=}\int_{−\infty} ^{\:\infty} \frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}+{e}^{{x}} \right)\left(\mathrm{1}+{e}^{−{x}} \right)}{dx} \\ $$

Question Number 123037    Answers: 5   Comments: 0

∫ ((√(1−x))/(1−(√x))) dx

$$\:\:\int\:\frac{\sqrt{\mathrm{1}−{x}}}{\mathrm{1}−\sqrt{{x}}}\:{dx} \\ $$$$ \\ $$

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