Prove $int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx=frac{sqrt{2}-2}{4} sqrt{pi}~ zeta left( frac{1}{2} right)$












8












$begingroup$


Wolfram Alpha evaluates this integral numerically as



$$int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx=0.379064 dots$$



Its value is apparently



$$frac{sqrt{2}-2}{4} sqrt{pi}~ zeta left( frac{1}{2} right)=0.37906401072dots$$




How would you solve this integral?




Obviously, we can make a substitution $t=x^2$



begin{align}
int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx&=frac{1}{2} int_0^{infty} frac{sqrt{t}}{cosh^2 (t)} dt\[10pt]
&=int_0^{infty} frac{sqrt{t}}{cosh (2t)+1} dt\[10pt]
&=frac{1}{2 sqrt{2}}int_0^{infty} frac{sqrt{u}}{cosh (u)+1} du
end{align}



We could use geometric series since $cosh (u) geq 1$, but I don't know how it will help.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
    $endgroup$
    – MrYouMath
    Jun 9 '16 at 22:01












  • $begingroup$
    It probably does, forgot about this, thank you
    $endgroup$
    – Yuriy S
    Jun 9 '16 at 22:03
















8












$begingroup$


Wolfram Alpha evaluates this integral numerically as



$$int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx=0.379064 dots$$



Its value is apparently



$$frac{sqrt{2}-2}{4} sqrt{pi}~ zeta left( frac{1}{2} right)=0.37906401072dots$$




How would you solve this integral?




Obviously, we can make a substitution $t=x^2$



begin{align}
int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx&=frac{1}{2} int_0^{infty} frac{sqrt{t}}{cosh^2 (t)} dt\[10pt]
&=int_0^{infty} frac{sqrt{t}}{cosh (2t)+1} dt\[10pt]
&=frac{1}{2 sqrt{2}}int_0^{infty} frac{sqrt{u}}{cosh (u)+1} du
end{align}



We could use geometric series since $cosh (u) geq 1$, but I don't know how it will help.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
    $endgroup$
    – MrYouMath
    Jun 9 '16 at 22:01












  • $begingroup$
    It probably does, forgot about this, thank you
    $endgroup$
    – Yuriy S
    Jun 9 '16 at 22:03














8












8








8


7



$begingroup$


Wolfram Alpha evaluates this integral numerically as



$$int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx=0.379064 dots$$



Its value is apparently



$$frac{sqrt{2}-2}{4} sqrt{pi}~ zeta left( frac{1}{2} right)=0.37906401072dots$$




How would you solve this integral?




Obviously, we can make a substitution $t=x^2$



begin{align}
int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx&=frac{1}{2} int_0^{infty} frac{sqrt{t}}{cosh^2 (t)} dt\[10pt]
&=int_0^{infty} frac{sqrt{t}}{cosh (2t)+1} dt\[10pt]
&=frac{1}{2 sqrt{2}}int_0^{infty} frac{sqrt{u}}{cosh (u)+1} du
end{align}



We could use geometric series since $cosh (u) geq 1$, but I don't know how it will help.










share|cite|improve this question











$endgroup$




Wolfram Alpha evaluates this integral numerically as



$$int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx=0.379064 dots$$



Its value is apparently



$$frac{sqrt{2}-2}{4} sqrt{pi}~ zeta left( frac{1}{2} right)=0.37906401072dots$$




How would you solve this integral?




Obviously, we can make a substitution $t=x^2$



begin{align}
int_0^{infty} frac{x^2}{cosh^2 (x^2)} dx&=frac{1}{2} int_0^{infty} frac{sqrt{t}}{cosh^2 (t)} dt\[10pt]
&=int_0^{infty} frac{sqrt{t}}{cosh (2t)+1} dt\[10pt]
&=frac{1}{2 sqrt{2}}int_0^{infty} frac{sqrt{u}}{cosh (u)+1} du
end{align}



We could use geometric series since $cosh (u) geq 1$, but I don't know how it will help.







calculus integration definite-integrals improper-integrals riemann-zeta






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edited Jun 10 '16 at 5:28









Sophie Agnesi

1,754324




1,754324










asked Jun 9 '16 at 21:57









Yuriy SYuriy S

15.8k433118




15.8k433118












  • $begingroup$
    Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
    $endgroup$
    – MrYouMath
    Jun 9 '16 at 22:01












  • $begingroup$
    It probably does, forgot about this, thank you
    $endgroup$
    – Yuriy S
    Jun 9 '16 at 22:03


















  • $begingroup$
    Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
    $endgroup$
    – MrYouMath
    Jun 9 '16 at 22:01












  • $begingroup$
    It probably does, forgot about this, thank you
    $endgroup$
    – Yuriy S
    Jun 9 '16 at 22:03
















$begingroup$
Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
$endgroup$
– MrYouMath
Jun 9 '16 at 22:01






$begingroup$
Maybe $Gamma (s) zeta (s)=int_0^{infty}frac{u^{s-1}}{e^u-1}du$ helps.
$endgroup$
– MrYouMath
Jun 9 '16 at 22:01














$begingroup$
It probably does, forgot about this, thank you
$endgroup$
– Yuriy S
Jun 9 '16 at 22:03




$begingroup$
It probably does, forgot about this, thank you
$endgroup$
– Yuriy S
Jun 9 '16 at 22:03










3 Answers
3






active

oldest

votes


















6












$begingroup$

$$I=frac{1}{2sqrt{2}}int_{0}^{+infty}frac{sqrt{u},du}{1+cosh(u)}=frac{1}{sqrt{2}}int_{1}^{+infty}frac{sqrt{log v}}{(v+1)^2},dv=frac{1}{sqrt{2}}int_{0}^{1}frac{sqrt{-log v}}{(1+v)^2},dv tag{1}$$
but since
$$ int_{0}^{1}v^k sqrt{-log v},dv = frac{sqrt{pi}}{2(1+k)^{3/2}} tag{2}$$
by expanding $frac{1}{(1+v)^2}$ as a Taylor series we get:




$$ I = frac{1}{sqrt{2}}sum_{ngeq 0}(-1)^n (n+1)frac{sqrt{pi}}{2(1+n)^{3/2}} = color{red}{frac{sqrt{pi}}{2sqrt{2}}cdotetaleft(frac{1}{2}right)}tag{3}$$




and the claim follows from the well-known:
$$ eta(s) = (1-2^{1-s}),zeta(s)tag{4} $$
that gives an analytic continuation for the $zeta$ function.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
    $endgroup$
    – Frank W.
    Jun 26 '18 at 23:26










  • $begingroup$
    @FrankW.: $u=log v$.
    $endgroup$
    – Jack D'Aurizio
    Jun 27 '18 at 12:18



















4












$begingroup$

Hint:



Consider the parametric integral



begin{equation}
I(a) = int_0^infty frac{t^{a-1}}{cosh^{2} t} dt=4 int_{0}^{infty} frac{t^{a-1}}{(e^{t}+e^{-t})^{2}} dt = 4 int_{0}^{infty}frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}} dt
end{equation}



Hence, your integral is simply



begin{equation}
int_0^{infty} frac{x^2}{cosh^2x^2} dx = -2 frac{partial}{partial b}left[int_0^{ infty}frac{t^{a-1}}{1+e^{bt}} dtright]_{a=frac{1}{2} , b=2}
end{equation}



I believe you can evaluate the last expression by your own.






share|cite|improve this answer











$endgroup$





















    3












    $begingroup$

    $newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
    newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
    newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
    newcommand{dd}{mathrm{d}}
    newcommand{ds}[1]{displaystyle{#1}}
    newcommand{expo}[1]{,mathrm{e}^{#1},}
    newcommand{half}{{1 over 2}}
    newcommand{ic}{mathrm{i}}
    newcommand{iff}{Leftrightarrow}
    newcommand{imp}{Longrightarrow}
    newcommand{ol}[1]{overline{#1}}
    newcommand{pars}[1]{left(, #1 ,right)}
    newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
    newcommand{root}[2]{,sqrt[#1]{, #2 ,},}
    newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
    newcommand{verts}[1]{leftvert, #1 ,rightvert}$



    begin{align}
    &bbox[10px,#ffd]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x},,
    stackrel{x to x^{1/2}}{=},,,
    2int_{0}^{infty}{x^{1/2}expo{2x} over pars{expo{2x} + 1}^{2}},dd x
    \[5mm] = &
    -int_{x = 0}^{x to infty}x^{1/2},ddpars{{1 over expo{2x} + 1}} =
    halfint_{0}^{infty}{x^{-1/2}expo{-2x} over 1 + expo{-2x}},dd x
    \[5mm] stackrel{2x to x}{=},,,&
    {root{2} over 4}int_{0}^{infty}{x^{-1/2}expo{-x} over 1 + expo{-x}},dd x
    \[5mm] = &
    {root{2} over 4}sum_{n = 0}^{infty}pars{-1}^{n}
    int_{0}^{infty}x^{-1/2}expo{-pars{n + 1}x},dd x
    \[3mm] stackrel{pars{n + 1}x to x}{=},,,&
    {root{2} over 4}sum_{n = 0}^{infty}
    {pars{-1}^{n} over pars{n + 1}^{1/2}} overbrace{%
    int_{0}^{infty}x^{-1/2}expo{-x},dd x}^{ds{Gammapars{half} = root{pi}}}
    \[5mm] = &
    {root{2} over 4},root{pi}sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{1/2}}
    end{align}




    With the identity
    $ds{sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{s}} =
    pars{1 - 2^{1 - s}}zetapars{s}}$
    :
    begin{align}
    bbox[#ffd,10px]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x} & =
    {root{2} over 4},root{pi}pars{1 - 2^{1 - 1/2}}zetapars{half}
    \[5mm] & =
    bbox[10px,border:1px groove navy]{{root{2} - 2 over 4},root{pi}zetapars{half}} approx 0.3791
    end{align}





    share|cite|improve this answer











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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      6












      $begingroup$

      $$I=frac{1}{2sqrt{2}}int_{0}^{+infty}frac{sqrt{u},du}{1+cosh(u)}=frac{1}{sqrt{2}}int_{1}^{+infty}frac{sqrt{log v}}{(v+1)^2},dv=frac{1}{sqrt{2}}int_{0}^{1}frac{sqrt{-log v}}{(1+v)^2},dv tag{1}$$
      but since
      $$ int_{0}^{1}v^k sqrt{-log v},dv = frac{sqrt{pi}}{2(1+k)^{3/2}} tag{2}$$
      by expanding $frac{1}{(1+v)^2}$ as a Taylor series we get:




      $$ I = frac{1}{sqrt{2}}sum_{ngeq 0}(-1)^n (n+1)frac{sqrt{pi}}{2(1+n)^{3/2}} = color{red}{frac{sqrt{pi}}{2sqrt{2}}cdotetaleft(frac{1}{2}right)}tag{3}$$




      and the claim follows from the well-known:
      $$ eta(s) = (1-2^{1-s}),zeta(s)tag{4} $$
      that gives an analytic continuation for the $zeta$ function.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
        $endgroup$
        – Frank W.
        Jun 26 '18 at 23:26










      • $begingroup$
        @FrankW.: $u=log v$.
        $endgroup$
        – Jack D'Aurizio
        Jun 27 '18 at 12:18
















      6












      $begingroup$

      $$I=frac{1}{2sqrt{2}}int_{0}^{+infty}frac{sqrt{u},du}{1+cosh(u)}=frac{1}{sqrt{2}}int_{1}^{+infty}frac{sqrt{log v}}{(v+1)^2},dv=frac{1}{sqrt{2}}int_{0}^{1}frac{sqrt{-log v}}{(1+v)^2},dv tag{1}$$
      but since
      $$ int_{0}^{1}v^k sqrt{-log v},dv = frac{sqrt{pi}}{2(1+k)^{3/2}} tag{2}$$
      by expanding $frac{1}{(1+v)^2}$ as a Taylor series we get:




      $$ I = frac{1}{sqrt{2}}sum_{ngeq 0}(-1)^n (n+1)frac{sqrt{pi}}{2(1+n)^{3/2}} = color{red}{frac{sqrt{pi}}{2sqrt{2}}cdotetaleft(frac{1}{2}right)}tag{3}$$




      and the claim follows from the well-known:
      $$ eta(s) = (1-2^{1-s}),zeta(s)tag{4} $$
      that gives an analytic continuation for the $zeta$ function.






      share|cite|improve this answer









      $endgroup$













      • $begingroup$
        Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
        $endgroup$
        – Frank W.
        Jun 26 '18 at 23:26










      • $begingroup$
        @FrankW.: $u=log v$.
        $endgroup$
        – Jack D'Aurizio
        Jun 27 '18 at 12:18














      6












      6








      6





      $begingroup$

      $$I=frac{1}{2sqrt{2}}int_{0}^{+infty}frac{sqrt{u},du}{1+cosh(u)}=frac{1}{sqrt{2}}int_{1}^{+infty}frac{sqrt{log v}}{(v+1)^2},dv=frac{1}{sqrt{2}}int_{0}^{1}frac{sqrt{-log v}}{(1+v)^2},dv tag{1}$$
      but since
      $$ int_{0}^{1}v^k sqrt{-log v},dv = frac{sqrt{pi}}{2(1+k)^{3/2}} tag{2}$$
      by expanding $frac{1}{(1+v)^2}$ as a Taylor series we get:




      $$ I = frac{1}{sqrt{2}}sum_{ngeq 0}(-1)^n (n+1)frac{sqrt{pi}}{2(1+n)^{3/2}} = color{red}{frac{sqrt{pi}}{2sqrt{2}}cdotetaleft(frac{1}{2}right)}tag{3}$$




      and the claim follows from the well-known:
      $$ eta(s) = (1-2^{1-s}),zeta(s)tag{4} $$
      that gives an analytic continuation for the $zeta$ function.






      share|cite|improve this answer









      $endgroup$



      $$I=frac{1}{2sqrt{2}}int_{0}^{+infty}frac{sqrt{u},du}{1+cosh(u)}=frac{1}{sqrt{2}}int_{1}^{+infty}frac{sqrt{log v}}{(v+1)^2},dv=frac{1}{sqrt{2}}int_{0}^{1}frac{sqrt{-log v}}{(1+v)^2},dv tag{1}$$
      but since
      $$ int_{0}^{1}v^k sqrt{-log v},dv = frac{sqrt{pi}}{2(1+k)^{3/2}} tag{2}$$
      by expanding $frac{1}{(1+v)^2}$ as a Taylor series we get:




      $$ I = frac{1}{sqrt{2}}sum_{ngeq 0}(-1)^n (n+1)frac{sqrt{pi}}{2(1+n)^{3/2}} = color{red}{frac{sqrt{pi}}{2sqrt{2}}cdotetaleft(frac{1}{2}right)}tag{3}$$




      and the claim follows from the well-known:
      $$ eta(s) = (1-2^{1-s}),zeta(s)tag{4} $$
      that gives an analytic continuation for the $zeta$ function.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Jun 9 '16 at 22:18









      Jack D'AurizioJack D'Aurizio

      1




      1












      • $begingroup$
        Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
        $endgroup$
        – Frank W.
        Jun 26 '18 at 23:26










      • $begingroup$
        @FrankW.: $u=log v$.
        $endgroup$
        – Jack D'Aurizio
        Jun 27 '18 at 12:18


















      • $begingroup$
        Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
        $endgroup$
        – Frank W.
        Jun 26 '18 at 23:26










      • $begingroup$
        @FrankW.: $u=log v$.
        $endgroup$
        – Jack D'Aurizio
        Jun 27 '18 at 12:18
















      $begingroup$
      Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
      $endgroup$
      – Frank W.
      Jun 26 '18 at 23:26




      $begingroup$
      Can you help me understand what substitution you made to get$$frac 1{2sqrt2}intlimits_0^{infty}du,frac {sqrt u}{cosh u+1}=frac 1{sqrt2}intlimits_1^{infty}dv,frac {sqrt{log v}}{(v+1)^2}$$?
      $endgroup$
      – Frank W.
      Jun 26 '18 at 23:26












      $begingroup$
      @FrankW.: $u=log v$.
      $endgroup$
      – Jack D'Aurizio
      Jun 27 '18 at 12:18




      $begingroup$
      @FrankW.: $u=log v$.
      $endgroup$
      – Jack D'Aurizio
      Jun 27 '18 at 12:18











      4












      $begingroup$

      Hint:



      Consider the parametric integral



      begin{equation}
      I(a) = int_0^infty frac{t^{a-1}}{cosh^{2} t} dt=4 int_{0}^{infty} frac{t^{a-1}}{(e^{t}+e^{-t})^{2}} dt = 4 int_{0}^{infty}frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}} dt
      end{equation}



      Hence, your integral is simply



      begin{equation}
      int_0^{infty} frac{x^2}{cosh^2x^2} dx = -2 frac{partial}{partial b}left[int_0^{ infty}frac{t^{a-1}}{1+e^{bt}} dtright]_{a=frac{1}{2} , b=2}
      end{equation}



      I believe you can evaluate the last expression by your own.






      share|cite|improve this answer











      $endgroup$


















        4












        $begingroup$

        Hint:



        Consider the parametric integral



        begin{equation}
        I(a) = int_0^infty frac{t^{a-1}}{cosh^{2} t} dt=4 int_{0}^{infty} frac{t^{a-1}}{(e^{t}+e^{-t})^{2}} dt = 4 int_{0}^{infty}frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}} dt
        end{equation}



        Hence, your integral is simply



        begin{equation}
        int_0^{infty} frac{x^2}{cosh^2x^2} dx = -2 frac{partial}{partial b}left[int_0^{ infty}frac{t^{a-1}}{1+e^{bt}} dtright]_{a=frac{1}{2} , b=2}
        end{equation}



        I believe you can evaluate the last expression by your own.






        share|cite|improve this answer











        $endgroup$
















          4












          4








          4





          $begingroup$

          Hint:



          Consider the parametric integral



          begin{equation}
          I(a) = int_0^infty frac{t^{a-1}}{cosh^{2} t} dt=4 int_{0}^{infty} frac{t^{a-1}}{(e^{t}+e^{-t})^{2}} dt = 4 int_{0}^{infty}frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}} dt
          end{equation}



          Hence, your integral is simply



          begin{equation}
          int_0^{infty} frac{x^2}{cosh^2x^2} dx = -2 frac{partial}{partial b}left[int_0^{ infty}frac{t^{a-1}}{1+e^{bt}} dtright]_{a=frac{1}{2} , b=2}
          end{equation}



          I believe you can evaluate the last expression by your own.






          share|cite|improve this answer











          $endgroup$



          Hint:



          Consider the parametric integral



          begin{equation}
          I(a) = int_0^infty frac{t^{a-1}}{cosh^{2} t} dt=4 int_{0}^{infty} frac{t^{a-1}}{(e^{t}+e^{-t})^{2}} dt = 4 int_{0}^{infty}frac{t^{a-1} e^{-2t}}{(1+e^{-2t})^{2}} dt
          end{equation}



          Hence, your integral is simply



          begin{equation}
          int_0^{infty} frac{x^2}{cosh^2x^2} dx = -2 frac{partial}{partial b}left[int_0^{ infty}frac{t^{a-1}}{1+e^{bt}} dtright]_{a=frac{1}{2} , b=2}
          end{equation}



          I believe you can evaluate the last expression by your own.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jun 10 '16 at 5:14

























          answered Jun 10 '16 at 5:06









          Sophie AgnesiSophie Agnesi

          1,754324




          1,754324























              3












              $begingroup$

              $newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
              newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
              newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
              newcommand{dd}{mathrm{d}}
              newcommand{ds}[1]{displaystyle{#1}}
              newcommand{expo}[1]{,mathrm{e}^{#1},}
              newcommand{half}{{1 over 2}}
              newcommand{ic}{mathrm{i}}
              newcommand{iff}{Leftrightarrow}
              newcommand{imp}{Longrightarrow}
              newcommand{ol}[1]{overline{#1}}
              newcommand{pars}[1]{left(, #1 ,right)}
              newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
              newcommand{root}[2]{,sqrt[#1]{, #2 ,},}
              newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
              newcommand{verts}[1]{leftvert, #1 ,rightvert}$



              begin{align}
              &bbox[10px,#ffd]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x},,
              stackrel{x to x^{1/2}}{=},,,
              2int_{0}^{infty}{x^{1/2}expo{2x} over pars{expo{2x} + 1}^{2}},dd x
              \[5mm] = &
              -int_{x = 0}^{x to infty}x^{1/2},ddpars{{1 over expo{2x} + 1}} =
              halfint_{0}^{infty}{x^{-1/2}expo{-2x} over 1 + expo{-2x}},dd x
              \[5mm] stackrel{2x to x}{=},,,&
              {root{2} over 4}int_{0}^{infty}{x^{-1/2}expo{-x} over 1 + expo{-x}},dd x
              \[5mm] = &
              {root{2} over 4}sum_{n = 0}^{infty}pars{-1}^{n}
              int_{0}^{infty}x^{-1/2}expo{-pars{n + 1}x},dd x
              \[3mm] stackrel{pars{n + 1}x to x}{=},,,&
              {root{2} over 4}sum_{n = 0}^{infty}
              {pars{-1}^{n} over pars{n + 1}^{1/2}} overbrace{%
              int_{0}^{infty}x^{-1/2}expo{-x},dd x}^{ds{Gammapars{half} = root{pi}}}
              \[5mm] = &
              {root{2} over 4},root{pi}sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{1/2}}
              end{align}




              With the identity
              $ds{sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{s}} =
              pars{1 - 2^{1 - s}}zetapars{s}}$
              :
              begin{align}
              bbox[#ffd,10px]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x} & =
              {root{2} over 4},root{pi}pars{1 - 2^{1 - 1/2}}zetapars{half}
              \[5mm] & =
              bbox[10px,border:1px groove navy]{{root{2} - 2 over 4},root{pi}zetapars{half}} approx 0.3791
              end{align}





              share|cite|improve this answer











              $endgroup$


















                3












                $begingroup$

                $newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
                newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
                newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
                newcommand{dd}{mathrm{d}}
                newcommand{ds}[1]{displaystyle{#1}}
                newcommand{expo}[1]{,mathrm{e}^{#1},}
                newcommand{half}{{1 over 2}}
                newcommand{ic}{mathrm{i}}
                newcommand{iff}{Leftrightarrow}
                newcommand{imp}{Longrightarrow}
                newcommand{ol}[1]{overline{#1}}
                newcommand{pars}[1]{left(, #1 ,right)}
                newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                newcommand{root}[2]{,sqrt[#1]{, #2 ,},}
                newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                newcommand{verts}[1]{leftvert, #1 ,rightvert}$



                begin{align}
                &bbox[10px,#ffd]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x},,
                stackrel{x to x^{1/2}}{=},,,
                2int_{0}^{infty}{x^{1/2}expo{2x} over pars{expo{2x} + 1}^{2}},dd x
                \[5mm] = &
                -int_{x = 0}^{x to infty}x^{1/2},ddpars{{1 over expo{2x} + 1}} =
                halfint_{0}^{infty}{x^{-1/2}expo{-2x} over 1 + expo{-2x}},dd x
                \[5mm] stackrel{2x to x}{=},,,&
                {root{2} over 4}int_{0}^{infty}{x^{-1/2}expo{-x} over 1 + expo{-x}},dd x
                \[5mm] = &
                {root{2} over 4}sum_{n = 0}^{infty}pars{-1}^{n}
                int_{0}^{infty}x^{-1/2}expo{-pars{n + 1}x},dd x
                \[3mm] stackrel{pars{n + 1}x to x}{=},,,&
                {root{2} over 4}sum_{n = 0}^{infty}
                {pars{-1}^{n} over pars{n + 1}^{1/2}} overbrace{%
                int_{0}^{infty}x^{-1/2}expo{-x},dd x}^{ds{Gammapars{half} = root{pi}}}
                \[5mm] = &
                {root{2} over 4},root{pi}sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{1/2}}
                end{align}




                With the identity
                $ds{sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{s}} =
                pars{1 - 2^{1 - s}}zetapars{s}}$
                :
                begin{align}
                bbox[#ffd,10px]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x} & =
                {root{2} over 4},root{pi}pars{1 - 2^{1 - 1/2}}zetapars{half}
                \[5mm] & =
                bbox[10px,border:1px groove navy]{{root{2} - 2 over 4},root{pi}zetapars{half}} approx 0.3791
                end{align}





                share|cite|improve this answer











                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  $newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
                  newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
                  newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
                  newcommand{dd}{mathrm{d}}
                  newcommand{ds}[1]{displaystyle{#1}}
                  newcommand{expo}[1]{,mathrm{e}^{#1},}
                  newcommand{half}{{1 over 2}}
                  newcommand{ic}{mathrm{i}}
                  newcommand{iff}{Leftrightarrow}
                  newcommand{imp}{Longrightarrow}
                  newcommand{ol}[1]{overline{#1}}
                  newcommand{pars}[1]{left(, #1 ,right)}
                  newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                  newcommand{root}[2]{,sqrt[#1]{, #2 ,},}
                  newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                  newcommand{verts}[1]{leftvert, #1 ,rightvert}$



                  begin{align}
                  &bbox[10px,#ffd]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x},,
                  stackrel{x to x^{1/2}}{=},,,
                  2int_{0}^{infty}{x^{1/2}expo{2x} over pars{expo{2x} + 1}^{2}},dd x
                  \[5mm] = &
                  -int_{x = 0}^{x to infty}x^{1/2},ddpars{{1 over expo{2x} + 1}} =
                  halfint_{0}^{infty}{x^{-1/2}expo{-2x} over 1 + expo{-2x}},dd x
                  \[5mm] stackrel{2x to x}{=},,,&
                  {root{2} over 4}int_{0}^{infty}{x^{-1/2}expo{-x} over 1 + expo{-x}},dd x
                  \[5mm] = &
                  {root{2} over 4}sum_{n = 0}^{infty}pars{-1}^{n}
                  int_{0}^{infty}x^{-1/2}expo{-pars{n + 1}x},dd x
                  \[3mm] stackrel{pars{n + 1}x to x}{=},,,&
                  {root{2} over 4}sum_{n = 0}^{infty}
                  {pars{-1}^{n} over pars{n + 1}^{1/2}} overbrace{%
                  int_{0}^{infty}x^{-1/2}expo{-x},dd x}^{ds{Gammapars{half} = root{pi}}}
                  \[5mm] = &
                  {root{2} over 4},root{pi}sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{1/2}}
                  end{align}




                  With the identity
                  $ds{sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{s}} =
                  pars{1 - 2^{1 - s}}zetapars{s}}$
                  :
                  begin{align}
                  bbox[#ffd,10px]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x} & =
                  {root{2} over 4},root{pi}pars{1 - 2^{1 - 1/2}}zetapars{half}
                  \[5mm] & =
                  bbox[10px,border:1px groove navy]{{root{2} - 2 over 4},root{pi}zetapars{half}} approx 0.3791
                  end{align}





                  share|cite|improve this answer











                  $endgroup$



                  $newcommand{angles}[1]{leftlangle, #1 ,rightrangle}
                  newcommand{braces}[1]{leftlbrace, #1 ,rightrbrace}
                  newcommand{bracks}[1]{leftlbrack, #1 ,rightrbrack}
                  newcommand{dd}{mathrm{d}}
                  newcommand{ds}[1]{displaystyle{#1}}
                  newcommand{expo}[1]{,mathrm{e}^{#1},}
                  newcommand{half}{{1 over 2}}
                  newcommand{ic}{mathrm{i}}
                  newcommand{iff}{Leftrightarrow}
                  newcommand{imp}{Longrightarrow}
                  newcommand{ol}[1]{overline{#1}}
                  newcommand{pars}[1]{left(, #1 ,right)}
                  newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                  newcommand{root}[2]{,sqrt[#1]{, #2 ,},}
                  newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                  newcommand{verts}[1]{leftvert, #1 ,rightvert}$



                  begin{align}
                  &bbox[10px,#ffd]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x},,
                  stackrel{x to x^{1/2}}{=},,,
                  2int_{0}^{infty}{x^{1/2}expo{2x} over pars{expo{2x} + 1}^{2}},dd x
                  \[5mm] = &
                  -int_{x = 0}^{x to infty}x^{1/2},ddpars{{1 over expo{2x} + 1}} =
                  halfint_{0}^{infty}{x^{-1/2}expo{-2x} over 1 + expo{-2x}},dd x
                  \[5mm] stackrel{2x to x}{=},,,&
                  {root{2} over 4}int_{0}^{infty}{x^{-1/2}expo{-x} over 1 + expo{-x}},dd x
                  \[5mm] = &
                  {root{2} over 4}sum_{n = 0}^{infty}pars{-1}^{n}
                  int_{0}^{infty}x^{-1/2}expo{-pars{n + 1}x},dd x
                  \[3mm] stackrel{pars{n + 1}x to x}{=},,,&
                  {root{2} over 4}sum_{n = 0}^{infty}
                  {pars{-1}^{n} over pars{n + 1}^{1/2}} overbrace{%
                  int_{0}^{infty}x^{-1/2}expo{-x},dd x}^{ds{Gammapars{half} = root{pi}}}
                  \[5mm] = &
                  {root{2} over 4},root{pi}sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{1/2}}
                  end{align}




                  With the identity
                  $ds{sum_{n = 1}^{infty}{pars{-1}^{n + 1} over n^{s}} =
                  pars{1 - 2^{1 - s}}zetapars{s}}$
                  :
                  begin{align}
                  bbox[#ffd,10px]{int_{0}^{infty}{x^{2} over cosh^{2}pars{x^{2}}},dd x} & =
                  {root{2} over 4},root{pi}pars{1 - 2^{1 - 1/2}}zetapars{half}
                  \[5mm] & =
                  bbox[10px,border:1px groove navy]{{root{2} - 2 over 4},root{pi}zetapars{half}} approx 0.3791
                  end{align}






                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 9 '18 at 0:38

























                  answered Jun 10 '16 at 4:13









                  Felix MarinFelix Marin

                  67.8k7107142




                  67.8k7107142






























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