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* Page found: Das ideale Fermigas (eq math.2547.29)

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\begin{align}

& \ln Y\approx \left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}{{p}^{2}}dp\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \\

& =\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \left[ \left. \left( \frac{{{p}^{3}}}{3}\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \right) \right|_{0}^{\infty }-\int_{0}^{\infty }{{}}{{\frac{p}{3}}^{3}}\frac{-\beta \frac{p}{m}\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}}}{\left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right)}dp \right] \\

& \left. \left( \frac{{{p}^{3}}}{3}\ln \left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right) \right) \right|_{0}^{\infty }=0 \\

& \Rightarrow \ln Y=-\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}{{\frac{p}{3}}^{3}}\frac{-\beta \frac{p}{m}\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}}}{\left( 1+\xi {{e}^{-\beta \frac{{{p}^{2}}}{2m}}} \right)}dp=\frac{2}{3}\left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}dp{{p}^{2}}\frac{\beta \frac{{{p}^{2}}}{2m}}{\left( \frac{1}{\xi }{{e}^{\beta \frac{{{p}^{2}}}{2m}}}+1 \right)} \\

& =\frac{2}{3}\beta \left( 2s+1 \right)\left( \frac{V}{{{h}^{3}}} \right)4\pi \int_{0}^{\infty }{{}}dp{{p}^{2}}\left\langle N(p) \right\rangle \frac{{{p}^{2}}}{2m} \\

\end{align}

TeX (checked): 

{\begin{aligned}&\ln Y\approx \left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}{{p}^{2}}dp\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\\&=\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \left[\left.\left({\frac {{p}^{3}}{3}}\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\right)\right|_{0}^{\infty }-\int _{0}^{\infty }{}{{\frac {p}{3}}^{3}}{\frac {-\beta {\frac {p}{m}}\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}}{\left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)}}dp\right]\\&\left.\left({\frac {{p}^{3}}{3}}\ln \left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)\right)\right|_{0}^{\infty }=0\\&\Rightarrow \ln Y=-\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}{{\frac {p}{3}}^{3}}{\frac {-\beta {\frac {p}{m}}\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}}{\left(1+\xi {{e}^{-\beta {\frac {{p}^{2}}{2m}}}}\right)}}dp={\frac {2}{3}}\left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}dp{{p}^{2}}{\frac {\beta {\frac {{p}^{2}}{2m}}}{\left({\frac {1}{\xi }}{{e}^{\beta {\frac {{p}^{2}}{2m}}}}+1\right)}}\\&={\frac {2}{3}}\beta \left(2s+1\right)\left({\frac {V}{{h}^{3}}}\right)4\pi \int _{0}^{\infty }{}dp{{p}^{2}}\left\langle N(p)\right\rangle {\frac {{p}^{2}}{2m}}\\\end{aligned}}

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ln ⁡ Y ≈ ( 2 ⁢ s + 1 ) ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ∫ 0 ∞ p 2 ⁢ 𝑑 p ⁢ ln ⁡ ( 1 + ξ ⁢ e - β ⁢ p 2 2 ⁢ m ) = ( 2 ⁢ s + 1 ) ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ [ ( p 3 3 ⁢ ln ⁡ ( 1 + ξ ⁢ e - β ⁢ p 2 2 ⁢ m ) ) | 0 ∞ - ∫ 0 ∞ p 3 3 ⁢ - β ⁢ p m ⁢ ξ ⁢ e - β ⁢ p 2 2 ⁢ m ( 1 + ξ ⁢ e - β ⁢ p 2 2 ⁢ m ) ⁢ 𝑑 p ] ( p 3 3 ⁢ ln ⁡ ( 1 + ξ ⁢ e - β ⁢ p 2 2 ⁢ m ) ) | 0 ∞ = 0 ⇒ ln ⁡ Y = - ( 2 ⁢ s + 1 ) ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ∫ 0 ∞ p 3 3 ⁢ - β ⁢ p m ⁢ ξ ⁢ e - β ⁢ p 2 2 ⁢ m ( 1 + ξ ⁢ e - β ⁢ p 2 2 ⁢ m ) ⁢ 𝑑 p = 2 3 ⁢ ( 2 ⁢ s + 1 ) ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ∫ 0 ∞ 𝑑 p ⁢ p 2 ⁢ β ⁢ p 2 2 ⁢ m ( 1 ξ ⁢ e β ⁢ p 2 2 ⁢ m + 1 ) = 2 3 ⁢ β ⁢ ( 2 ⁢ s + 1 ) ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ∫ 0 ∞ 𝑑 p ⁢ p 2 ⁢ ⟨ N ⁢ ( p ) ⟩ ⁢ p 2 2 ⁢ m absent 𝑌 2 𝑠 1 𝑉 superscript ℎ 3 4 𝜋 superscript subscript 0 superscript 𝑝 2 differential-d 𝑝 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 absent absent 2 𝑠 1 𝑉 superscript ℎ 3 4 𝜋 delimited-[] evaluated-at superscript 𝑝 3 3 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 0 superscript subscript 0 superscript 𝑝 3 3 𝛽 𝑝 𝑚 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 differential-d 𝑝 absent evaluated-at superscript 𝑝 3 3 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 0 0 absent ⇒ absent 𝑌 2 𝑠 1 𝑉 superscript ℎ 3 4 𝜋 superscript subscript 0 superscript 𝑝 3 3 𝛽 𝑝 𝑚 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 differential-d 𝑝 2 3 2 𝑠 1 𝑉 superscript ℎ 3 4 𝜋 superscript subscript 0 differential-d 𝑝 superscript 𝑝 2 𝛽 superscript 𝑝 2 2 𝑚 1 𝜉 superscript 𝑒 𝛽 superscript 𝑝 2 2 𝑚 1 absent absent 2 3 𝛽 2 𝑠 1 𝑉 superscript ℎ 3 4 𝜋 superscript subscript 0 differential-d 𝑝 superscript 𝑝 2 delimited-⟨⟩ 𝑁 𝑝 superscript 𝑝 2 2 𝑚 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\ln Y\approx% \left(2s+1\right)\left(\frac{V}{{{h}^{3}}}\right)4\pi\int_{0}^{\infty}{{}}{{p}% ^{2}}dp\ln\left(1+\xi{{e}^{-\beta\frac{{{p}^{2}}}{2m}}}\right)\\ \par&\displaystyle=\left(2s+1\right)\left(\frac{V}{{{h}^{3}}}\right)4\pi\left[% \left.\left(\frac{{{p}^{3}}}{3}\ln\left(1+\xi{{e}^{-\beta\frac{{{p}^{2}}}{2m}}% }\right)\right)\right|_{0}^{\infty}-\int_{0}^{\infty}{{}}{{\frac{p}{3}}^{3}}% \frac{-\beta\frac{p}{m}\xi{{e}^{-\beta\frac{{{p}^{2}}}{2m}}}}{\left(1+\xi{{e}^% {-\beta\frac{{{p}^{2}}}{2m}}}\right)}dp\right]\\ \par&\displaystyle\left.\left(\frac{{{p}^{3}}}{3}\ln\left(1+\xi{{e}^{-\beta% \frac{{{p}^{2}}}{2m}}}\right)\right)\right|_{0}^{\infty}=0\\ \par&\displaystyle\Rightarrow\ln Y=-\left(2s+1\right)\left(\frac{V}{{{h}^{3}}}% \right)4\pi\int_{0}^{\infty}{{}}{{\frac{p}{3}}^{3}}\frac{-\beta\frac{p}{m}\xi{% {e}^{-\beta\frac{{{p}^{2}}}{2m}}}}{\left(1+\xi{{e}^{-\beta\frac{{{p}^{2}}}{2m}% }}\right)}dp=\frac{2}{3}\left(2s+1\right)\left(\frac{V}{{{h}^{3}}}\right)4\pi% \int_{0}^{\infty}{{}}dp{{p}^{2}}\frac{\beta\frac{{{p}^{2}}}{2m}}{\left(\frac{1% }{\xi}{{e}^{\beta\frac{{{p}^{2}}}{2m}}}+1\right)}\\ \par&\displaystyle=\frac{2}{3}\beta\left(2s+1\right)\left(\frac{V}{{{h}^{3}}}% \right)4\pi\int_{0}^{\infty}{{}}dp{{p}^{2}}\left\langle N(p)\right\rangle\frac% {{{p}^{2}}}{2m}\\ \par\end{aligned}}}

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ln⁡Y≈(2s+1)(Vh3)4π∫0∞p2dpln⁡(1+ξe−βp22m)=(2s+1)(Vh3)4π[(p33ln⁡(1+ξe−βp22m))|0∞−∫0∞p33−βpmξe−βp22m(1+ξe−βp22m)dp](p33ln⁡(1+ξe−βp22m))|0∞=0⇒ln⁡Y=−(2s+1)(Vh3)4π∫0∞p33−βpmξe−βp22m(1+ξe−βp22m)dp=23(2s+1)(Vh3)4π∫0∞dpp2βp22m(1ξeβp22m+1)=23β(2s+1)(Vh3)4π∫0∞dpp2⟨N(p)⟩p22m

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