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sdvanced-cslculus-if-x-R-and-x-0-x-e-t-1-t-ln-x-t-dt-then-prove-that-0-e-x-x-dx-2-

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Question Number 136381 by mnjuly1970 last updated on 21/Mar/21
......sdvanced cslculus...... if x∈R^+ and:: 𝛗(x)=∫_0 ^( x) ((e^t −1)/t)ln((x/t))dt then prove that :: Ψ=∫_0 ^( ∞) e^(−x) 𝛗(x)dx=ζ(2)
……sdvancedcslculus……ifx∈R+and::ϕ(x)=∫0xet−1tln(xt)dtthenprovethat::Ψ=∫0∞e−xϕ(x)dx=ζ(2)
Answered by Ñï= last updated on 21/Mar/21
Φ(x)=∫_0 ^x ((e^t −1)/t)ln((x/t))dt =∫_0 ^1 ((1−e^(xt) )/t)lntdt =∫_0 ^1 ((1−e^(x(1−t)) )/(1−t))ln(1−t)dt Ψ=∫_0 ^∞ e^(−x) ∫_0 ^1 ((1−e^(x(1−t)) )/(1−t))ln(1−t)dtdx =∫_0 ^∞ ∫_0 ^1 ((e^(−x) −e^(−xt) )/(1−t))ln(1−t)dtdx =∫_0 ^1 ((1−(1/t))/(1−t))ln(1−t)dt =−∫_0 ^1 ((ln(1−t))/t)dt =Li_2 (1)
Φ(x)=∫0xet−1tln(xt)dt=∫011−exttlntdt=∫011−ex(1−t)1−tln(1−t)dtΨ=∫0∞e−x∫011−ex(1−t)1−tln(1−t)dtdx=∫0∞∫01e−x−e−xt1−tln(1−t)dtdx=∫011−1t1−tln(1−t)dt=−∫01ln(1−t)tdt=Li2(1)
Commented by mnjuly1970 last updated on 21/Mar/21
thank you sir...grateful..
thankyousir…grateful..

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