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Question Number 123793    Answers: 2   Comments: 0

∫ ((4x−1)/(2x^2 −3x+2)) dx ?

$$\:\:\int\:\frac{\mathrm{4}{x}−\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}\:{dx}\:? \\ $$

Question Number 123769    Answers: 0   Comments: 1

Question Number 123768    Answers: 0   Comments: 0

Question Number 123767    Answers: 1   Comments: 0

Question Number 123764    Answers: 1   Comments: 0

.... advanced calculus ... prove that:::: Σ_(n=1) ^∞ {((ζ(2n+1))/(4^(n ) (2n+1)))}=ln(2)−γ γ:: euler−mascheroni constant

$$\:\:\:\:\:\:\:\:\:\:\:\:....\:{advanced}\:\:{calculus}\:... \\ $$$$\:\:\:\:\:\:\:{prove}\:\:{that}:::: \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{\frac{\zeta\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{4}^{{n}\:} \:\left(\mathrm{2}{n}+\mathrm{1}\right)}\right\}={ln}\left(\mathrm{2}\right)−\gamma \\ $$$$\:\:\:\:\:\:\:\:\gamma::\:\:{euler}−{mascheroni} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{constant} \\ $$

Question Number 123745    Answers: 0   Comments: 0

...advanced calculus... evaluation of: Ω=∫_0 ^( (π/2)) sin(x)log(sin(x))dx by using the euler beta and gamma function: β(p,(1/2))=2∫_0 ^(π/2) sin^(2p−1) (x)dx ((dβ(p,(1/2)))/dp) =2∫_0 ^(π/2) 2sin^(2p−1) (x)ln(sin(x))dx =4∫_0 ^(π/2) sin^(2p−1) (x)ln(sin(x))dx Ω=(1/4)[((dβ(p,(1/2)))/dp)]_(p=1) ((d(β(p,(1/2))))/dp)=(√π)[((Γ′(p)Γ(p+(1/2))−Γ′(p+(1/2))Γ(p))/(Γ^2 (p+(1/2))))]_(p=1) Ω=(1/4)((√π)[((Γ^′ (1)Γ((3/2))−Γ′((3/2))Γ(1))/(Γ^2 ((3/2))))]) =((√π)/4)[((−γ(((√π)/2))−((√π)/2)(2−γ−2ln(2)))/(π/4))] =((√π)/4)∗((√π)/2)(((−γ−2+γ+2ln(2))/(π/4))) =ln(2)−1 ... m.n.july.1970...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:\:{calculus}... \\ $$$$\:\:{evaluation}\:\:{of}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}\left({x}\right){log}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:{by}\:{using}\:{the}\:{euler}\:{beta}\:{and}\:{gamma}\:{function}: \\ $$$$\:\:\:\:\:\:\:\:\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{p}−\mathrm{1}} \left({x}\right){dx} \\ $$$$\:\:\:\frac{{d}\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)}{{dp}}\:=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{2}{sin}^{\mathrm{2}{p}−\mathrm{1}} \left({x}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$=\mathrm{4}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{p}−\mathrm{1}} \left({x}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\Omega=\frac{\mathrm{1}}{\mathrm{4}}\left[\frac{{d}\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)}{{dp}}\right]_{{p}=\mathrm{1}} \\ $$$$\:\frac{{d}\left(\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)\right)}{{dp}}=\sqrt{\pi}\left[\frac{\Gamma'\left({p}\right)\Gamma\left({p}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\Gamma'\left({p}+\frac{\mathrm{1}}{\mathrm{2}}\right)\Gamma\left({p}\right)}{\Gamma^{\mathrm{2}} \left({p}+\frac{\mathrm{1}}{\mathrm{2}}\right)}\right]_{{p}=\mathrm{1}} \\ $$$$\:\Omega=\frac{\mathrm{1}}{\mathrm{4}}\left(\sqrt{\pi}\left[\frac{\Gamma^{'} \left(\mathrm{1}\right)\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)−\Gamma'\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\Gamma\left(\mathrm{1}\right)}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)}\right]\right) \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}}\left[\frac{−\gamma\left(\frac{\sqrt{\pi}}{\mathrm{2}}\right)−\frac{\sqrt{\pi}}{\mathrm{2}}\left(\mathrm{2}−\gamma−\mathrm{2}{ln}\left(\mathrm{2}\right)\right)}{\frac{\pi}{\mathrm{4}}}\right] \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}}\ast\frac{\sqrt{\pi}}{\mathrm{2}}\left(\frac{−\gamma−\mathrm{2}+\gamma+\mathrm{2}{ln}\left(\mathrm{2}\right)}{\frac{\pi}{\mathrm{4}}}\right) \\ $$$$={ln}\left(\mathrm{2}\right)−\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:...\:{m}.{n}.{july}.\mathrm{1970}... \\ $$$$\:\:\: \\ $$

Question Number 123710    Answers: 2   Comments: 0

calculate ∫_1 ^(√3) (dx/((x^2 +1)^2 (x+2)^5 ))

$${calculate}\:\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} } \\ $$

Question Number 123703    Answers: 1   Comments: 0

Question Number 123676    Answers: 2   Comments: 0

find ∫_0 ^∞ e^(−x) ln(1+e^(2x) )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{2x}} \right)\mathrm{dx} \\ $$

Question Number 123674    Answers: 2   Comments: 0

find ∫_0 ^∞ e^(−2x) ln(1+e^x )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\mathrm{e}^{−\mathrm{2x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx} \\ $$

Question Number 123673    Answers: 0   Comments: 0

find ∫_0 ^∞ ((√(x^2 +x+1))−(√(x^2 +x)))^4 dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\:\left(\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}−\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}\right)^{\mathrm{4}} \mathrm{dx} \\ $$

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

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

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