integral $ int_0^{frac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}(x+frac{pi}{3})}right)...












1












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We want to evaluate



$ displaystyle int_0^{frac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}(x+frac{pi}{3})}right) mathrm{d}x$.



We tried contour integration which was not helpful. Then we started trying to use symmetries, this also doesn't work.



Our tries










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    $begingroup$
    This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
    $endgroup$
    – Ron Gordon
    Feb 25 '18 at 2:30










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    Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
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    – GNUSupporter 8964民主女神 地下教會
    Dec 13 '18 at 9:44
















1












$begingroup$


We want to evaluate



$ displaystyle int_0^{frac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}(x+frac{pi}{3})}right) mathrm{d}x$.



We tried contour integration which was not helpful. Then we started trying to use symmetries, this also doesn't work.



Our tries










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
    $endgroup$
    – Ron Gordon
    Feb 25 '18 at 2:30










  • $begingroup$
    Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 13 '18 at 9:44














1












1








1


1



$begingroup$


We want to evaluate



$ displaystyle int_0^{frac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}(x+frac{pi}{3})}right) mathrm{d}x$.



We tried contour integration which was not helpful. Then we started trying to use symmetries, this also doesn't work.



Our tries










share|cite|improve this question











$endgroup$




We want to evaluate



$ displaystyle int_0^{frac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}(x+frac{pi}{3})}right) mathrm{d}x$.



We tried contour integration which was not helpful. Then we started trying to use symmetries, this also doesn't work.



Our tries







real-analysis integration complex-analysis limits






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edited Feb 24 '18 at 23:37









Arjang

5,59162363




5,59162363










asked Feb 24 '18 at 21:51









JanJan

478215




478215








  • 1




    $begingroup$
    This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
    $endgroup$
    – Ron Gordon
    Feb 25 '18 at 2:30










  • $begingroup$
    Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 13 '18 at 9:44














  • 1




    $begingroup$
    This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
    $endgroup$
    – Ron Gordon
    Feb 25 '18 at 2:30










  • $begingroup$
    Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
    $endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Dec 13 '18 at 9:44








1




1




$begingroup$
This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
$endgroup$
– Ron Gordon
Feb 25 '18 at 2:30




$begingroup$
This is not exactly the same thing, but I think it will help: math.stackexchange.com/questions/305124/…
$endgroup$
– Ron Gordon
Feb 25 '18 at 2:30












$begingroup$
Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Dec 13 '18 at 9:44




$begingroup$
Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
Dec 13 '18 at 9:44










5 Answers
5






active

oldest

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5












$begingroup$

Here is an approach that uses real methods only.



begin{align*}
I &= int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx\
&= int_0^{pi/3} ln (sin x) , dx - int_0^{pi/3} ln left (sin left (x + frac{pi}{3} right ) right ) , dx.
end{align*}
If in the second integral appearing on the right a substitution of $x mapsto x - frac{pi}{3}$ is enforced, one has
begin{align*}
I &= int_0^{pi/3} ln (sin x) , dx - int_{pi/3}^{2pi/3} ln (sin x) , dx\
&= 2 int_0^{pi/3} ln (sin x) , dx - int_0^{2pi/3} ln (sin x) , dx.tag1
end{align*}



Now consider the integral
$$I(alpha) = int_0^alpha ln (sin x) , dx, quad 0 < alpha < pi.tag2$$
Taking advantage of the well-known identity
$$ln (sin x) = -ln 2 - sum_{k = 1}^infty frac{cos (2kx)}{k}, quad 0 < x < pi,$$
substituting this result into (2), after interchanging the summation with the integration before integrating one finds
$$I(alpha) = -alpha ln 2 - frac{1}{2} sum_{k = 1}^infty frac{sin (2k alpha)}{k^2} = -alpha ln 2 - frac{1}{2} text{Cl}_2 (alpha).$$
Here $text{Cl}_2 (varphi)$ denotes the Clausen function of order two.



In terms of the Clausen function of order two the integral in (1) can be written as
$$I = -text{Cl}_2 left (frac{2pi}{3} right ) + frac{1}{2} text{Cl}_2 left (frac{4pi}{3} right ).tag3$$
From the duplication formula for the Clausen function of order two, namely
$$text{Cl}_2 (2theta) = 2 text{Cl}_2 (theta) - 2 text{Cl}_2 (pi - theta), quad 0 < theta < pi,$$
if we set $theta = 2pi/3$ in the above duplication formula, as
$$text{Cl}_2 left (frac{4pi}{3} right ) = 2 text{Cl}_2 left (frac{2pi}{3} right ) - 2 text{Cl}_2 left (frac{pi}{3} right ),$$
the expression for our integral in (3) can be expressed more simply as
$$int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx = -text{Cl}_2 left (frac{pi}{3} right ).$$






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    $begingroup$

    Hint:



    As shown in this answer,
    $$
    log(sin(x))=-log(2)-sum_{k=1}^inftyfrac{cos(2kx)}{k}
    $$
    Therefore,
    $$
    intlog(sin(x)),mathrm{d}x=-xlog(2)-sum_{k=1}^inftyfrac{sin(2kx)}{2k^2}
    $$





    Applying the hint:
    $$
    begin{align}
    int_0^{pi/3}logleft(frac{sin(x)}{sin(x+pi/3)}right),mathrm{d}x
    &=int_0^{pi/3}log(sin(x)),mathrm{d}x-int_{pi/3}^{2pi/3}log(sin(x)),mathrm{d}x\
    &=-2sum_{k=1}^inftyfrac{sin(2kpi/3)}{2k^2}+sum_{k=1}^inftyfrac{sin(4kpi/3)}{2k^2}\
    &=-sum_{k=1}^inftyfrac{sin(2kpi/3)}{k^2}+2sum_{k=1}^inftyfrac{sin(4kpi/3)}{4k^2}\
    &=sum_{k=1}^infty(-1)^kfrac{sin(2kpi/3)}{k^2}\
    &=-sum_{k=1}^inftyfrac{sin(kpi/3)}{k^2}\
    &=-frac{sqrt3}2sum_{k=0}^infty(-1)^kleft(frac1{(3k+1)^2}+frac1{(3k+2)^2}right)
    end{align}
    $$






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      3












      $begingroup$

      $newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
      newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
      newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
      newcommand{dd}{mathrm{d}}
      newcommand{ds}[1]{displaystyle{#1}}
      newcommand{expo}[1]{,mathrm{e}^{#1},}
      newcommand{ic}{mathrm{i}}
      newcommand{mc}[1]{mathcal{#1}}
      newcommand{mrm}[1]{mathrm{#1}}
      newcommand{pars}[1]{left(,{#1},right)}
      newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
      newcommand{root}[2]{,sqrt[#1]{,{#2},},}
      newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
      newcommand{verts}[1]{leftvert,{#1},rightvert}$
      begin{align}
      &bbox[10px,#ffd]{ds{%
      int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
      int_{-pi/6}^{pi/6}lnpars{sinpars{x + pi/6} over cospars{x}},dd x
      \[5mm] = &
      int_{-pi/6}^{pi/6}lnpars{{root{3} over 2},tanpars{x} + { 1 over 2}}
      ,dd x
      ,,,,,,stackrel{large x = arctanpars{2t - 1 over root{3}}}{=},,,
      ,,,
      {root{3} over 2}int_{0}^{1}{lnpars{t} over t^{2} - t + 1},dd t
      \[5mm] = &
      {root{3} over 2}int_{0}^{1}{lnpars{t} over pars{t - r}pars{t - bar{r}}},dd tlabel{1}tag{1}
      end{align}




      where $ds{r equiv {1 over 2} + {root{3} over 2},ic = exppars{{pi over 3},ic}}$




      eqref{1} is reduced to
      begin{align}
      &bbox[10px,#ffd]{ds{%
      int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
      {root{3} over 2}int_{0}^{1}lnpars{t}
      pars{{1 over t - r} - {1 over t - bar{r}}}{1 over r - bar{r}},dd t
      \[5mm] = &
      {root{3} over 2},{1 over 2ic,Impars{r}},2ic,Imint_{0}^{1}{lnpars{t} over t - r},dd t =
      -,Imint_{0}^{1}{lnpars{t} over r - t},dd t =
      -,Imint_{0}^{1/r}{lnpars{rt} over 1 - t},dd t
      \[5mm] = &
      -,Imint_{0}^{largebar{r}}{lnpars{1 - t} over t},dd t =
      Imint_{0}^{largebar{r}}mrm{Li}_{2}'pars{t},dd tqquad
      pars{~mrm{Li}_{s} mbox{is the} PolyLogarithm Function~}
      \[5mm] implies &
      bbx{int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x = Immrm{Li}_{2}pars{expo{-piic/3}} approx -1.0149}
      end{align}






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        1












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        The given integral can be approached as follows:
        $$begin{eqnarray*} mathcal{I}=int_{0}^{pi/3}logleft(frac{sin x}{frac{1}{2}sin x+frac{sqrt{3}}{2}cos x}right),dx&=&frac{pi}{3}log 2-int_{0}^{pi/3}log(1+sqrt{3}cottheta),dtheta\&=&frac{pi}{3}log 2-int_{1/sqrt{3}}^{+infty}frac{log(1+tsqrt{3})}{1+t^2},dt\&=&frac{pi}
        {3}log2-int_{0}^{sqrt{3}}frac{logleft(1+frac{sqrt{3}}{t}right)}{1+t^2},dt\&=&frac{pi}{3}log2-sqrt{3}int_{1}^{+infty}frac{log(z+1)}{3+z^2},dz\(text{trigamma})qquad&=&-frac{1}{24sqrt{3}}left(psi'left(tfrac{1}{6}right)+psi'left(tfrac{1}{3}right)-psi'left(tfrac{2}{3}right)-psi'left(tfrac{5}{6}right)right)end{eqnarray*} $$
        from which it follows that
        $$mathcal{I}=-frac{sqrt{3}}{2}sum_{kgeq 0}left(frac{1}{(6k+1)^2}+frac{1}{(6k+2)^2}-frac{1}{(6k+4)^2}-frac{1}{(6k+5)^2}right).tag{1}$$
        We don't have a really nicer closed form, but the terms of the last series behaves like $frac{1}{18k^3}$ for large values of $k$, ensuring a decent speed of convergence of the partial sums. Due to the reflection formulas for the trigamma function we have
        $$sum_{kgeq 0}left(frac{1}{(6k+1)^2}color{red}{+}frac{1}{(6k+5)^2}right)=frac{pi^2}{9},qquad sum_{kgeq 0}left(frac{1}{(6k+2)^2}color{red}{+}frac{1}{(6k+4)^2}right)=frac{pi^2}{27} $$
        which can be used to devise acceleration tricks, even if the signs pattern in the previous identity is the "wrong" one. Numerically $mathcal{I}approx -1.0149416$.



        $(1)$ can be proved also by applying termwise integration to the Fourier series of $logsin(x)$, which is well-known.






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          1












          $begingroup$

          Hint: $displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x=displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}({mathrm{sin}(x)})dx-int_0^{dfrac{pi}{3}} lnleft({mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x$






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          $endgroup$













          • $begingroup$
            This is a hint towards what?
            $endgroup$
            – Jack D'Aurizio
            Feb 25 '18 at 17:41










          • $begingroup$
            @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
            $endgroup$
            – Mehrdad Zandigohar
            Feb 25 '18 at 18:05










          • $begingroup$
            I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
            $endgroup$
            – Jack D'Aurizio
            Feb 25 '18 at 18:08












          • $begingroup$
            I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
            $endgroup$
            – Mehrdad Zandigohar
            Feb 25 '18 at 18:14










          • $begingroup$
            Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
            $endgroup$
            – Jack D'Aurizio
            Feb 25 '18 at 18:16













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          5 Answers
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          5 Answers
          5






          active

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          active

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          active

          oldest

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          5












          $begingroup$

          Here is an approach that uses real methods only.



          begin{align*}
          I &= int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx\
          &= int_0^{pi/3} ln (sin x) , dx - int_0^{pi/3} ln left (sin left (x + frac{pi}{3} right ) right ) , dx.
          end{align*}
          If in the second integral appearing on the right a substitution of $x mapsto x - frac{pi}{3}$ is enforced, one has
          begin{align*}
          I &= int_0^{pi/3} ln (sin x) , dx - int_{pi/3}^{2pi/3} ln (sin x) , dx\
          &= 2 int_0^{pi/3} ln (sin x) , dx - int_0^{2pi/3} ln (sin x) , dx.tag1
          end{align*}



          Now consider the integral
          $$I(alpha) = int_0^alpha ln (sin x) , dx, quad 0 < alpha < pi.tag2$$
          Taking advantage of the well-known identity
          $$ln (sin x) = -ln 2 - sum_{k = 1}^infty frac{cos (2kx)}{k}, quad 0 < x < pi,$$
          substituting this result into (2), after interchanging the summation with the integration before integrating one finds
          $$I(alpha) = -alpha ln 2 - frac{1}{2} sum_{k = 1}^infty frac{sin (2k alpha)}{k^2} = -alpha ln 2 - frac{1}{2} text{Cl}_2 (alpha).$$
          Here $text{Cl}_2 (varphi)$ denotes the Clausen function of order two.



          In terms of the Clausen function of order two the integral in (1) can be written as
          $$I = -text{Cl}_2 left (frac{2pi}{3} right ) + frac{1}{2} text{Cl}_2 left (frac{4pi}{3} right ).tag3$$
          From the duplication formula for the Clausen function of order two, namely
          $$text{Cl}_2 (2theta) = 2 text{Cl}_2 (theta) - 2 text{Cl}_2 (pi - theta), quad 0 < theta < pi,$$
          if we set $theta = 2pi/3$ in the above duplication formula, as
          $$text{Cl}_2 left (frac{4pi}{3} right ) = 2 text{Cl}_2 left (frac{2pi}{3} right ) - 2 text{Cl}_2 left (frac{pi}{3} right ),$$
          the expression for our integral in (3) can be expressed more simply as
          $$int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx = -text{Cl}_2 left (frac{pi}{3} right ).$$






          share|cite|improve this answer











          $endgroup$


















            5












            $begingroup$

            Here is an approach that uses real methods only.



            begin{align*}
            I &= int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx\
            &= int_0^{pi/3} ln (sin x) , dx - int_0^{pi/3} ln left (sin left (x + frac{pi}{3} right ) right ) , dx.
            end{align*}
            If in the second integral appearing on the right a substitution of $x mapsto x - frac{pi}{3}$ is enforced, one has
            begin{align*}
            I &= int_0^{pi/3} ln (sin x) , dx - int_{pi/3}^{2pi/3} ln (sin x) , dx\
            &= 2 int_0^{pi/3} ln (sin x) , dx - int_0^{2pi/3} ln (sin x) , dx.tag1
            end{align*}



            Now consider the integral
            $$I(alpha) = int_0^alpha ln (sin x) , dx, quad 0 < alpha < pi.tag2$$
            Taking advantage of the well-known identity
            $$ln (sin x) = -ln 2 - sum_{k = 1}^infty frac{cos (2kx)}{k}, quad 0 < x < pi,$$
            substituting this result into (2), after interchanging the summation with the integration before integrating one finds
            $$I(alpha) = -alpha ln 2 - frac{1}{2} sum_{k = 1}^infty frac{sin (2k alpha)}{k^2} = -alpha ln 2 - frac{1}{2} text{Cl}_2 (alpha).$$
            Here $text{Cl}_2 (varphi)$ denotes the Clausen function of order two.



            In terms of the Clausen function of order two the integral in (1) can be written as
            $$I = -text{Cl}_2 left (frac{2pi}{3} right ) + frac{1}{2} text{Cl}_2 left (frac{4pi}{3} right ).tag3$$
            From the duplication formula for the Clausen function of order two, namely
            $$text{Cl}_2 (2theta) = 2 text{Cl}_2 (theta) - 2 text{Cl}_2 (pi - theta), quad 0 < theta < pi,$$
            if we set $theta = 2pi/3$ in the above duplication formula, as
            $$text{Cl}_2 left (frac{4pi}{3} right ) = 2 text{Cl}_2 left (frac{2pi}{3} right ) - 2 text{Cl}_2 left (frac{pi}{3} right ),$$
            the expression for our integral in (3) can be expressed more simply as
            $$int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx = -text{Cl}_2 left (frac{pi}{3} right ).$$






            share|cite|improve this answer











            $endgroup$
















              5












              5








              5





              $begingroup$

              Here is an approach that uses real methods only.



              begin{align*}
              I &= int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx\
              &= int_0^{pi/3} ln (sin x) , dx - int_0^{pi/3} ln left (sin left (x + frac{pi}{3} right ) right ) , dx.
              end{align*}
              If in the second integral appearing on the right a substitution of $x mapsto x - frac{pi}{3}$ is enforced, one has
              begin{align*}
              I &= int_0^{pi/3} ln (sin x) , dx - int_{pi/3}^{2pi/3} ln (sin x) , dx\
              &= 2 int_0^{pi/3} ln (sin x) , dx - int_0^{2pi/3} ln (sin x) , dx.tag1
              end{align*}



              Now consider the integral
              $$I(alpha) = int_0^alpha ln (sin x) , dx, quad 0 < alpha < pi.tag2$$
              Taking advantage of the well-known identity
              $$ln (sin x) = -ln 2 - sum_{k = 1}^infty frac{cos (2kx)}{k}, quad 0 < x < pi,$$
              substituting this result into (2), after interchanging the summation with the integration before integrating one finds
              $$I(alpha) = -alpha ln 2 - frac{1}{2} sum_{k = 1}^infty frac{sin (2k alpha)}{k^2} = -alpha ln 2 - frac{1}{2} text{Cl}_2 (alpha).$$
              Here $text{Cl}_2 (varphi)$ denotes the Clausen function of order two.



              In terms of the Clausen function of order two the integral in (1) can be written as
              $$I = -text{Cl}_2 left (frac{2pi}{3} right ) + frac{1}{2} text{Cl}_2 left (frac{4pi}{3} right ).tag3$$
              From the duplication formula for the Clausen function of order two, namely
              $$text{Cl}_2 (2theta) = 2 text{Cl}_2 (theta) - 2 text{Cl}_2 (pi - theta), quad 0 < theta < pi,$$
              if we set $theta = 2pi/3$ in the above duplication formula, as
              $$text{Cl}_2 left (frac{4pi}{3} right ) = 2 text{Cl}_2 left (frac{2pi}{3} right ) - 2 text{Cl}_2 left (frac{pi}{3} right ),$$
              the expression for our integral in (3) can be expressed more simply as
              $$int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx = -text{Cl}_2 left (frac{pi}{3} right ).$$






              share|cite|improve this answer











              $endgroup$



              Here is an approach that uses real methods only.



              begin{align*}
              I &= int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx\
              &= int_0^{pi/3} ln (sin x) , dx - int_0^{pi/3} ln left (sin left (x + frac{pi}{3} right ) right ) , dx.
              end{align*}
              If in the second integral appearing on the right a substitution of $x mapsto x - frac{pi}{3}$ is enforced, one has
              begin{align*}
              I &= int_0^{pi/3} ln (sin x) , dx - int_{pi/3}^{2pi/3} ln (sin x) , dx\
              &= 2 int_0^{pi/3} ln (sin x) , dx - int_0^{2pi/3} ln (sin x) , dx.tag1
              end{align*}



              Now consider the integral
              $$I(alpha) = int_0^alpha ln (sin x) , dx, quad 0 < alpha < pi.tag2$$
              Taking advantage of the well-known identity
              $$ln (sin x) = -ln 2 - sum_{k = 1}^infty frac{cos (2kx)}{k}, quad 0 < x < pi,$$
              substituting this result into (2), after interchanging the summation with the integration before integrating one finds
              $$I(alpha) = -alpha ln 2 - frac{1}{2} sum_{k = 1}^infty frac{sin (2k alpha)}{k^2} = -alpha ln 2 - frac{1}{2} text{Cl}_2 (alpha).$$
              Here $text{Cl}_2 (varphi)$ denotes the Clausen function of order two.



              In terms of the Clausen function of order two the integral in (1) can be written as
              $$I = -text{Cl}_2 left (frac{2pi}{3} right ) + frac{1}{2} text{Cl}_2 left (frac{4pi}{3} right ).tag3$$
              From the duplication formula for the Clausen function of order two, namely
              $$text{Cl}_2 (2theta) = 2 text{Cl}_2 (theta) - 2 text{Cl}_2 (pi - theta), quad 0 < theta < pi,$$
              if we set $theta = 2pi/3$ in the above duplication formula, as
              $$text{Cl}_2 left (frac{4pi}{3} right ) = 2 text{Cl}_2 left (frac{2pi}{3} right ) - 2 text{Cl}_2 left (frac{pi}{3} right ),$$
              the expression for our integral in (3) can be expressed more simply as
              $$int_0^{pi/3} ln left (frac{sin x}{sin left (x + frac{pi}{3} right )} right ) , dx = -text{Cl}_2 left (frac{pi}{3} right ).$$







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              edited Feb 25 '18 at 0:06

























              answered Feb 24 '18 at 23:33









              omegadotomegadot

              5,5802728




              5,5802728























                  3












                  $begingroup$

                  Hint:



                  As shown in this answer,
                  $$
                  log(sin(x))=-log(2)-sum_{k=1}^inftyfrac{cos(2kx)}{k}
                  $$
                  Therefore,
                  $$
                  intlog(sin(x)),mathrm{d}x=-xlog(2)-sum_{k=1}^inftyfrac{sin(2kx)}{2k^2}
                  $$





                  Applying the hint:
                  $$
                  begin{align}
                  int_0^{pi/3}logleft(frac{sin(x)}{sin(x+pi/3)}right),mathrm{d}x
                  &=int_0^{pi/3}log(sin(x)),mathrm{d}x-int_{pi/3}^{2pi/3}log(sin(x)),mathrm{d}x\
                  &=-2sum_{k=1}^inftyfrac{sin(2kpi/3)}{2k^2}+sum_{k=1}^inftyfrac{sin(4kpi/3)}{2k^2}\
                  &=-sum_{k=1}^inftyfrac{sin(2kpi/3)}{k^2}+2sum_{k=1}^inftyfrac{sin(4kpi/3)}{4k^2}\
                  &=sum_{k=1}^infty(-1)^kfrac{sin(2kpi/3)}{k^2}\
                  &=-sum_{k=1}^inftyfrac{sin(kpi/3)}{k^2}\
                  &=-frac{sqrt3}2sum_{k=0}^infty(-1)^kleft(frac1{(3k+1)^2}+frac1{(3k+2)^2}right)
                  end{align}
                  $$






                  share|cite|improve this answer











                  $endgroup$


















                    3












                    $begingroup$

                    Hint:



                    As shown in this answer,
                    $$
                    log(sin(x))=-log(2)-sum_{k=1}^inftyfrac{cos(2kx)}{k}
                    $$
                    Therefore,
                    $$
                    intlog(sin(x)),mathrm{d}x=-xlog(2)-sum_{k=1}^inftyfrac{sin(2kx)}{2k^2}
                    $$





                    Applying the hint:
                    $$
                    begin{align}
                    int_0^{pi/3}logleft(frac{sin(x)}{sin(x+pi/3)}right),mathrm{d}x
                    &=int_0^{pi/3}log(sin(x)),mathrm{d}x-int_{pi/3}^{2pi/3}log(sin(x)),mathrm{d}x\
                    &=-2sum_{k=1}^inftyfrac{sin(2kpi/3)}{2k^2}+sum_{k=1}^inftyfrac{sin(4kpi/3)}{2k^2}\
                    &=-sum_{k=1}^inftyfrac{sin(2kpi/3)}{k^2}+2sum_{k=1}^inftyfrac{sin(4kpi/3)}{4k^2}\
                    &=sum_{k=1}^infty(-1)^kfrac{sin(2kpi/3)}{k^2}\
                    &=-sum_{k=1}^inftyfrac{sin(kpi/3)}{k^2}\
                    &=-frac{sqrt3}2sum_{k=0}^infty(-1)^kleft(frac1{(3k+1)^2}+frac1{(3k+2)^2}right)
                    end{align}
                    $$






                    share|cite|improve this answer











                    $endgroup$
















                      3












                      3








                      3





                      $begingroup$

                      Hint:



                      As shown in this answer,
                      $$
                      log(sin(x))=-log(2)-sum_{k=1}^inftyfrac{cos(2kx)}{k}
                      $$
                      Therefore,
                      $$
                      intlog(sin(x)),mathrm{d}x=-xlog(2)-sum_{k=1}^inftyfrac{sin(2kx)}{2k^2}
                      $$





                      Applying the hint:
                      $$
                      begin{align}
                      int_0^{pi/3}logleft(frac{sin(x)}{sin(x+pi/3)}right),mathrm{d}x
                      &=int_0^{pi/3}log(sin(x)),mathrm{d}x-int_{pi/3}^{2pi/3}log(sin(x)),mathrm{d}x\
                      &=-2sum_{k=1}^inftyfrac{sin(2kpi/3)}{2k^2}+sum_{k=1}^inftyfrac{sin(4kpi/3)}{2k^2}\
                      &=-sum_{k=1}^inftyfrac{sin(2kpi/3)}{k^2}+2sum_{k=1}^inftyfrac{sin(4kpi/3)}{4k^2}\
                      &=sum_{k=1}^infty(-1)^kfrac{sin(2kpi/3)}{k^2}\
                      &=-sum_{k=1}^inftyfrac{sin(kpi/3)}{k^2}\
                      &=-frac{sqrt3}2sum_{k=0}^infty(-1)^kleft(frac1{(3k+1)^2}+frac1{(3k+2)^2}right)
                      end{align}
                      $$






                      share|cite|improve this answer











                      $endgroup$



                      Hint:



                      As shown in this answer,
                      $$
                      log(sin(x))=-log(2)-sum_{k=1}^inftyfrac{cos(2kx)}{k}
                      $$
                      Therefore,
                      $$
                      intlog(sin(x)),mathrm{d}x=-xlog(2)-sum_{k=1}^inftyfrac{sin(2kx)}{2k^2}
                      $$





                      Applying the hint:
                      $$
                      begin{align}
                      int_0^{pi/3}logleft(frac{sin(x)}{sin(x+pi/3)}right),mathrm{d}x
                      &=int_0^{pi/3}log(sin(x)),mathrm{d}x-int_{pi/3}^{2pi/3}log(sin(x)),mathrm{d}x\
                      &=-2sum_{k=1}^inftyfrac{sin(2kpi/3)}{2k^2}+sum_{k=1}^inftyfrac{sin(4kpi/3)}{2k^2}\
                      &=-sum_{k=1}^inftyfrac{sin(2kpi/3)}{k^2}+2sum_{k=1}^inftyfrac{sin(4kpi/3)}{4k^2}\
                      &=sum_{k=1}^infty(-1)^kfrac{sin(2kpi/3)}{k^2}\
                      &=-sum_{k=1}^inftyfrac{sin(kpi/3)}{k^2}\
                      &=-frac{sqrt3}2sum_{k=0}^infty(-1)^kleft(frac1{(3k+1)^2}+frac1{(3k+2)^2}right)
                      end{align}
                      $$







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                      edited Feb 25 '18 at 21:40

























                      answered Feb 25 '18 at 20:47









                      robjohnrobjohn

                      267k27308631




                      267k27308631























                          3












                          $begingroup$

                          $newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
                          newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
                          newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
                          newcommand{dd}{mathrm{d}}
                          newcommand{ds}[1]{displaystyle{#1}}
                          newcommand{expo}[1]{,mathrm{e}^{#1},}
                          newcommand{ic}{mathrm{i}}
                          newcommand{mc}[1]{mathcal{#1}}
                          newcommand{mrm}[1]{mathrm{#1}}
                          newcommand{pars}[1]{left(,{#1},right)}
                          newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                          newcommand{root}[2]{,sqrt[#1]{,{#2},},}
                          newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                          newcommand{verts}[1]{leftvert,{#1},rightvert}$
                          begin{align}
                          &bbox[10px,#ffd]{ds{%
                          int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                          int_{-pi/6}^{pi/6}lnpars{sinpars{x + pi/6} over cospars{x}},dd x
                          \[5mm] = &
                          int_{-pi/6}^{pi/6}lnpars{{root{3} over 2},tanpars{x} + { 1 over 2}}
                          ,dd x
                          ,,,,,,stackrel{large x = arctanpars{2t - 1 over root{3}}}{=},,,
                          ,,,
                          {root{3} over 2}int_{0}^{1}{lnpars{t} over t^{2} - t + 1},dd t
                          \[5mm] = &
                          {root{3} over 2}int_{0}^{1}{lnpars{t} over pars{t - r}pars{t - bar{r}}},dd tlabel{1}tag{1}
                          end{align}




                          where $ds{r equiv {1 over 2} + {root{3} over 2},ic = exppars{{pi over 3},ic}}$




                          eqref{1} is reduced to
                          begin{align}
                          &bbox[10px,#ffd]{ds{%
                          int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                          {root{3} over 2}int_{0}^{1}lnpars{t}
                          pars{{1 over t - r} - {1 over t - bar{r}}}{1 over r - bar{r}},dd t
                          \[5mm] = &
                          {root{3} over 2},{1 over 2ic,Impars{r}},2ic,Imint_{0}^{1}{lnpars{t} over t - r},dd t =
                          -,Imint_{0}^{1}{lnpars{t} over r - t},dd t =
                          -,Imint_{0}^{1/r}{lnpars{rt} over 1 - t},dd t
                          \[5mm] = &
                          -,Imint_{0}^{largebar{r}}{lnpars{1 - t} over t},dd t =
                          Imint_{0}^{largebar{r}}mrm{Li}_{2}'pars{t},dd tqquad
                          pars{~mrm{Li}_{s} mbox{is the} PolyLogarithm Function~}
                          \[5mm] implies &
                          bbx{int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x = Immrm{Li}_{2}pars{expo{-piic/3}} approx -1.0149}
                          end{align}






                          share|cite|improve this answer











                          $endgroup$


















                            3












                            $begingroup$

                            $newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
                            newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
                            newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
                            newcommand{dd}{mathrm{d}}
                            newcommand{ds}[1]{displaystyle{#1}}
                            newcommand{expo}[1]{,mathrm{e}^{#1},}
                            newcommand{ic}{mathrm{i}}
                            newcommand{mc}[1]{mathcal{#1}}
                            newcommand{mrm}[1]{mathrm{#1}}
                            newcommand{pars}[1]{left(,{#1},right)}
                            newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                            newcommand{root}[2]{,sqrt[#1]{,{#2},},}
                            newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                            newcommand{verts}[1]{leftvert,{#1},rightvert}$
                            begin{align}
                            &bbox[10px,#ffd]{ds{%
                            int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                            int_{-pi/6}^{pi/6}lnpars{sinpars{x + pi/6} over cospars{x}},dd x
                            \[5mm] = &
                            int_{-pi/6}^{pi/6}lnpars{{root{3} over 2},tanpars{x} + { 1 over 2}}
                            ,dd x
                            ,,,,,,stackrel{large x = arctanpars{2t - 1 over root{3}}}{=},,,
                            ,,,
                            {root{3} over 2}int_{0}^{1}{lnpars{t} over t^{2} - t + 1},dd t
                            \[5mm] = &
                            {root{3} over 2}int_{0}^{1}{lnpars{t} over pars{t - r}pars{t - bar{r}}},dd tlabel{1}tag{1}
                            end{align}




                            where $ds{r equiv {1 over 2} + {root{3} over 2},ic = exppars{{pi over 3},ic}}$




                            eqref{1} is reduced to
                            begin{align}
                            &bbox[10px,#ffd]{ds{%
                            int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                            {root{3} over 2}int_{0}^{1}lnpars{t}
                            pars{{1 over t - r} - {1 over t - bar{r}}}{1 over r - bar{r}},dd t
                            \[5mm] = &
                            {root{3} over 2},{1 over 2ic,Impars{r}},2ic,Imint_{0}^{1}{lnpars{t} over t - r},dd t =
                            -,Imint_{0}^{1}{lnpars{t} over r - t},dd t =
                            -,Imint_{0}^{1/r}{lnpars{rt} over 1 - t},dd t
                            \[5mm] = &
                            -,Imint_{0}^{largebar{r}}{lnpars{1 - t} over t},dd t =
                            Imint_{0}^{largebar{r}}mrm{Li}_{2}'pars{t},dd tqquad
                            pars{~mrm{Li}_{s} mbox{is the} PolyLogarithm Function~}
                            \[5mm] implies &
                            bbx{int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x = Immrm{Li}_{2}pars{expo{-piic/3}} approx -1.0149}
                            end{align}






                            share|cite|improve this answer











                            $endgroup$
















                              3












                              3








                              3





                              $begingroup$

                              $newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
                              newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
                              newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
                              newcommand{dd}{mathrm{d}}
                              newcommand{ds}[1]{displaystyle{#1}}
                              newcommand{expo}[1]{,mathrm{e}^{#1},}
                              newcommand{ic}{mathrm{i}}
                              newcommand{mc}[1]{mathcal{#1}}
                              newcommand{mrm}[1]{mathrm{#1}}
                              newcommand{pars}[1]{left(,{#1},right)}
                              newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                              newcommand{root}[2]{,sqrt[#1]{,{#2},},}
                              newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                              newcommand{verts}[1]{leftvert,{#1},rightvert}$
                              begin{align}
                              &bbox[10px,#ffd]{ds{%
                              int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                              int_{-pi/6}^{pi/6}lnpars{sinpars{x + pi/6} over cospars{x}},dd x
                              \[5mm] = &
                              int_{-pi/6}^{pi/6}lnpars{{root{3} over 2},tanpars{x} + { 1 over 2}}
                              ,dd x
                              ,,,,,,stackrel{large x = arctanpars{2t - 1 over root{3}}}{=},,,
                              ,,,
                              {root{3} over 2}int_{0}^{1}{lnpars{t} over t^{2} - t + 1},dd t
                              \[5mm] = &
                              {root{3} over 2}int_{0}^{1}{lnpars{t} over pars{t - r}pars{t - bar{r}}},dd tlabel{1}tag{1}
                              end{align}




                              where $ds{r equiv {1 over 2} + {root{3} over 2},ic = exppars{{pi over 3},ic}}$




                              eqref{1} is reduced to
                              begin{align}
                              &bbox[10px,#ffd]{ds{%
                              int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                              {root{3} over 2}int_{0}^{1}lnpars{t}
                              pars{{1 over t - r} - {1 over t - bar{r}}}{1 over r - bar{r}},dd t
                              \[5mm] = &
                              {root{3} over 2},{1 over 2ic,Impars{r}},2ic,Imint_{0}^{1}{lnpars{t} over t - r},dd t =
                              -,Imint_{0}^{1}{lnpars{t} over r - t},dd t =
                              -,Imint_{0}^{1/r}{lnpars{rt} over 1 - t},dd t
                              \[5mm] = &
                              -,Imint_{0}^{largebar{r}}{lnpars{1 - t} over t},dd t =
                              Imint_{0}^{largebar{r}}mrm{Li}_{2}'pars{t},dd tqquad
                              pars{~mrm{Li}_{s} mbox{is the} PolyLogarithm Function~}
                              \[5mm] implies &
                              bbx{int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x = Immrm{Li}_{2}pars{expo{-piic/3}} approx -1.0149}
                              end{align}






                              share|cite|improve this answer











                              $endgroup$



                              $newcommand{bbx}[1]{,bbox[15px,border:1px groove navy]{displaystyle{#1}},}
                              newcommand{braces}[1]{leftlbrace,{#1},rightrbrace}
                              newcommand{bracks}[1]{leftlbrack,{#1},rightrbrack}
                              newcommand{dd}{mathrm{d}}
                              newcommand{ds}[1]{displaystyle{#1}}
                              newcommand{expo}[1]{,mathrm{e}^{#1},}
                              newcommand{ic}{mathrm{i}}
                              newcommand{mc}[1]{mathcal{#1}}
                              newcommand{mrm}[1]{mathrm{#1}}
                              newcommand{pars}[1]{left(,{#1},right)}
                              newcommand{partiald}[3]{frac{partial^{#1} #2}{partial #3^{#1}}}
                              newcommand{root}[2]{,sqrt[#1]{,{#2},},}
                              newcommand{totald}[3]{frac{mathrm{d}^{#1} #2}{mathrm{d} #3^{#1}}}
                              newcommand{verts}[1]{leftvert,{#1},rightvert}$
                              begin{align}
                              &bbox[10px,#ffd]{ds{%
                              int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                              int_{-pi/6}^{pi/6}lnpars{sinpars{x + pi/6} over cospars{x}},dd x
                              \[5mm] = &
                              int_{-pi/6}^{pi/6}lnpars{{root{3} over 2},tanpars{x} + { 1 over 2}}
                              ,dd x
                              ,,,,,,stackrel{large x = arctanpars{2t - 1 over root{3}}}{=},,,
                              ,,,
                              {root{3} over 2}int_{0}^{1}{lnpars{t} over t^{2} - t + 1},dd t
                              \[5mm] = &
                              {root{3} over 2}int_{0}^{1}{lnpars{t} over pars{t - r}pars{t - bar{r}}},dd tlabel{1}tag{1}
                              end{align}




                              where $ds{r equiv {1 over 2} + {root{3} over 2},ic = exppars{{pi over 3},ic}}$




                              eqref{1} is reduced to
                              begin{align}
                              &bbox[10px,#ffd]{ds{%
                              int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x}} =
                              {root{3} over 2}int_{0}^{1}lnpars{t}
                              pars{{1 over t - r} - {1 over t - bar{r}}}{1 over r - bar{r}},dd t
                              \[5mm] = &
                              {root{3} over 2},{1 over 2ic,Impars{r}},2ic,Imint_{0}^{1}{lnpars{t} over t - r},dd t =
                              -,Imint_{0}^{1}{lnpars{t} over r - t},dd t =
                              -,Imint_{0}^{1/r}{lnpars{rt} over 1 - t},dd t
                              \[5mm] = &
                              -,Imint_{0}^{largebar{r}}{lnpars{1 - t} over t},dd t =
                              Imint_{0}^{largebar{r}}mrm{Li}_{2}'pars{t},dd tqquad
                              pars{~mrm{Li}_{s} mbox{is the} PolyLogarithm Function~}
                              \[5mm] implies &
                              bbx{int_{0}^{pi/3}lnpars{sinpars{x} over sinpars{x + pi/3}},dd x = Immrm{Li}_{2}pars{expo{-piic/3}} approx -1.0149}
                              end{align}







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                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Feb 26 '18 at 22:32

























                              answered Feb 26 '18 at 0:32









                              Felix MarinFelix Marin

                              67.8k7107142




                              67.8k7107142























                                  1












                                  $begingroup$

                                  The given integral can be approached as follows:
                                  $$begin{eqnarray*} mathcal{I}=int_{0}^{pi/3}logleft(frac{sin x}{frac{1}{2}sin x+frac{sqrt{3}}{2}cos x}right),dx&=&frac{pi}{3}log 2-int_{0}^{pi/3}log(1+sqrt{3}cottheta),dtheta\&=&frac{pi}{3}log 2-int_{1/sqrt{3}}^{+infty}frac{log(1+tsqrt{3})}{1+t^2},dt\&=&frac{pi}
                                  {3}log2-int_{0}^{sqrt{3}}frac{logleft(1+frac{sqrt{3}}{t}right)}{1+t^2},dt\&=&frac{pi}{3}log2-sqrt{3}int_{1}^{+infty}frac{log(z+1)}{3+z^2},dz\(text{trigamma})qquad&=&-frac{1}{24sqrt{3}}left(psi'left(tfrac{1}{6}right)+psi'left(tfrac{1}{3}right)-psi'left(tfrac{2}{3}right)-psi'left(tfrac{5}{6}right)right)end{eqnarray*} $$
                                  from which it follows that
                                  $$mathcal{I}=-frac{sqrt{3}}{2}sum_{kgeq 0}left(frac{1}{(6k+1)^2}+frac{1}{(6k+2)^2}-frac{1}{(6k+4)^2}-frac{1}{(6k+5)^2}right).tag{1}$$
                                  We don't have a really nicer closed form, but the terms of the last series behaves like $frac{1}{18k^3}$ for large values of $k$, ensuring a decent speed of convergence of the partial sums. Due to the reflection formulas for the trigamma function we have
                                  $$sum_{kgeq 0}left(frac{1}{(6k+1)^2}color{red}{+}frac{1}{(6k+5)^2}right)=frac{pi^2}{9},qquad sum_{kgeq 0}left(frac{1}{(6k+2)^2}color{red}{+}frac{1}{(6k+4)^2}right)=frac{pi^2}{27} $$
                                  which can be used to devise acceleration tricks, even if the signs pattern in the previous identity is the "wrong" one. Numerically $mathcal{I}approx -1.0149416$.



                                  $(1)$ can be proved also by applying termwise integration to the Fourier series of $logsin(x)$, which is well-known.






                                  share|cite|improve this answer











                                  $endgroup$


















                                    1












                                    $begingroup$

                                    The given integral can be approached as follows:
                                    $$begin{eqnarray*} mathcal{I}=int_{0}^{pi/3}logleft(frac{sin x}{frac{1}{2}sin x+frac{sqrt{3}}{2}cos x}right),dx&=&frac{pi}{3}log 2-int_{0}^{pi/3}log(1+sqrt{3}cottheta),dtheta\&=&frac{pi}{3}log 2-int_{1/sqrt{3}}^{+infty}frac{log(1+tsqrt{3})}{1+t^2},dt\&=&frac{pi}
                                    {3}log2-int_{0}^{sqrt{3}}frac{logleft(1+frac{sqrt{3}}{t}right)}{1+t^2},dt\&=&frac{pi}{3}log2-sqrt{3}int_{1}^{+infty}frac{log(z+1)}{3+z^2},dz\(text{trigamma})qquad&=&-frac{1}{24sqrt{3}}left(psi'left(tfrac{1}{6}right)+psi'left(tfrac{1}{3}right)-psi'left(tfrac{2}{3}right)-psi'left(tfrac{5}{6}right)right)end{eqnarray*} $$
                                    from which it follows that
                                    $$mathcal{I}=-frac{sqrt{3}}{2}sum_{kgeq 0}left(frac{1}{(6k+1)^2}+frac{1}{(6k+2)^2}-frac{1}{(6k+4)^2}-frac{1}{(6k+5)^2}right).tag{1}$$
                                    We don't have a really nicer closed form, but the terms of the last series behaves like $frac{1}{18k^3}$ for large values of $k$, ensuring a decent speed of convergence of the partial sums. Due to the reflection formulas for the trigamma function we have
                                    $$sum_{kgeq 0}left(frac{1}{(6k+1)^2}color{red}{+}frac{1}{(6k+5)^2}right)=frac{pi^2}{9},qquad sum_{kgeq 0}left(frac{1}{(6k+2)^2}color{red}{+}frac{1}{(6k+4)^2}right)=frac{pi^2}{27} $$
                                    which can be used to devise acceleration tricks, even if the signs pattern in the previous identity is the "wrong" one. Numerically $mathcal{I}approx -1.0149416$.



                                    $(1)$ can be proved also by applying termwise integration to the Fourier series of $logsin(x)$, which is well-known.






                                    share|cite|improve this answer











                                    $endgroup$
















                                      1












                                      1








                                      1





                                      $begingroup$

                                      The given integral can be approached as follows:
                                      $$begin{eqnarray*} mathcal{I}=int_{0}^{pi/3}logleft(frac{sin x}{frac{1}{2}sin x+frac{sqrt{3}}{2}cos x}right),dx&=&frac{pi}{3}log 2-int_{0}^{pi/3}log(1+sqrt{3}cottheta),dtheta\&=&frac{pi}{3}log 2-int_{1/sqrt{3}}^{+infty}frac{log(1+tsqrt{3})}{1+t^2},dt\&=&frac{pi}
                                      {3}log2-int_{0}^{sqrt{3}}frac{logleft(1+frac{sqrt{3}}{t}right)}{1+t^2},dt\&=&frac{pi}{3}log2-sqrt{3}int_{1}^{+infty}frac{log(z+1)}{3+z^2},dz\(text{trigamma})qquad&=&-frac{1}{24sqrt{3}}left(psi'left(tfrac{1}{6}right)+psi'left(tfrac{1}{3}right)-psi'left(tfrac{2}{3}right)-psi'left(tfrac{5}{6}right)right)end{eqnarray*} $$
                                      from which it follows that
                                      $$mathcal{I}=-frac{sqrt{3}}{2}sum_{kgeq 0}left(frac{1}{(6k+1)^2}+frac{1}{(6k+2)^2}-frac{1}{(6k+4)^2}-frac{1}{(6k+5)^2}right).tag{1}$$
                                      We don't have a really nicer closed form, but the terms of the last series behaves like $frac{1}{18k^3}$ for large values of $k$, ensuring a decent speed of convergence of the partial sums. Due to the reflection formulas for the trigamma function we have
                                      $$sum_{kgeq 0}left(frac{1}{(6k+1)^2}color{red}{+}frac{1}{(6k+5)^2}right)=frac{pi^2}{9},qquad sum_{kgeq 0}left(frac{1}{(6k+2)^2}color{red}{+}frac{1}{(6k+4)^2}right)=frac{pi^2}{27} $$
                                      which can be used to devise acceleration tricks, even if the signs pattern in the previous identity is the "wrong" one. Numerically $mathcal{I}approx -1.0149416$.



                                      $(1)$ can be proved also by applying termwise integration to the Fourier series of $logsin(x)$, which is well-known.






                                      share|cite|improve this answer











                                      $endgroup$



                                      The given integral can be approached as follows:
                                      $$begin{eqnarray*} mathcal{I}=int_{0}^{pi/3}logleft(frac{sin x}{frac{1}{2}sin x+frac{sqrt{3}}{2}cos x}right),dx&=&frac{pi}{3}log 2-int_{0}^{pi/3}log(1+sqrt{3}cottheta),dtheta\&=&frac{pi}{3}log 2-int_{1/sqrt{3}}^{+infty}frac{log(1+tsqrt{3})}{1+t^2},dt\&=&frac{pi}
                                      {3}log2-int_{0}^{sqrt{3}}frac{logleft(1+frac{sqrt{3}}{t}right)}{1+t^2},dt\&=&frac{pi}{3}log2-sqrt{3}int_{1}^{+infty}frac{log(z+1)}{3+z^2},dz\(text{trigamma})qquad&=&-frac{1}{24sqrt{3}}left(psi'left(tfrac{1}{6}right)+psi'left(tfrac{1}{3}right)-psi'left(tfrac{2}{3}right)-psi'left(tfrac{5}{6}right)right)end{eqnarray*} $$
                                      from which it follows that
                                      $$mathcal{I}=-frac{sqrt{3}}{2}sum_{kgeq 0}left(frac{1}{(6k+1)^2}+frac{1}{(6k+2)^2}-frac{1}{(6k+4)^2}-frac{1}{(6k+5)^2}right).tag{1}$$
                                      We don't have a really nicer closed form, but the terms of the last series behaves like $frac{1}{18k^3}$ for large values of $k$, ensuring a decent speed of convergence of the partial sums. Due to the reflection formulas for the trigamma function we have
                                      $$sum_{kgeq 0}left(frac{1}{(6k+1)^2}color{red}{+}frac{1}{(6k+5)^2}right)=frac{pi^2}{9},qquad sum_{kgeq 0}left(frac{1}{(6k+2)^2}color{red}{+}frac{1}{(6k+4)^2}right)=frac{pi^2}{27} $$
                                      which can be used to devise acceleration tricks, even if the signs pattern in the previous identity is the "wrong" one. Numerically $mathcal{I}approx -1.0149416$.



                                      $(1)$ can be proved also by applying termwise integration to the Fourier series of $logsin(x)$, which is well-known.







                                      share|cite|improve this answer














                                      share|cite|improve this answer



                                      share|cite|improve this answer








                                      edited Feb 25 '18 at 18:49

























                                      answered Feb 25 '18 at 17:37









                                      Jack D'AurizioJack D'Aurizio

                                      289k33281661




                                      289k33281661























                                          1












                                          $begingroup$

                                          Hint: $displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x=displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}({mathrm{sin}(x)})dx-int_0^{dfrac{pi}{3}} lnleft({mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x$






                                          share|cite|improve this answer











                                          $endgroup$













                                          • $begingroup$
                                            This is a hint towards what?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 17:41










                                          • $begingroup$
                                            @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:05










                                          • $begingroup$
                                            I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:08












                                          • $begingroup$
                                            I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:14










                                          • $begingroup$
                                            Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:16


















                                          1












                                          $begingroup$

                                          Hint: $displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x=displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}({mathrm{sin}(x)})dx-int_0^{dfrac{pi}{3}} lnleft({mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x$






                                          share|cite|improve this answer











                                          $endgroup$













                                          • $begingroup$
                                            This is a hint towards what?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 17:41










                                          • $begingroup$
                                            @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:05










                                          • $begingroup$
                                            I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:08












                                          • $begingroup$
                                            I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:14










                                          • $begingroup$
                                            Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:16
















                                          1












                                          1








                                          1





                                          $begingroup$

                                          Hint: $displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x=displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}({mathrm{sin}(x)})dx-int_0^{dfrac{pi}{3}} lnleft({mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x$






                                          share|cite|improve this answer











                                          $endgroup$



                                          Hint: $displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}left(frac{mathrm{sin}(x)}{mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x=displaystyle int_0^{dfrac{pi}{3}}mathrm{ln}({mathrm{sin}(x)})dx-int_0^{dfrac{pi}{3}} lnleft({mathrm{sin}left(x+frac{pi}{3}right)}right) mathrm{d}x$







                                          share|cite|improve this answer














                                          share|cite|improve this answer



                                          share|cite|improve this answer








                                          edited Dec 13 '18 at 9:43









                                          DavidG

                                          2,0511722




                                          2,0511722










                                          answered Feb 24 '18 at 22:01









                                          Mehrdad ZandigoharMehrdad Zandigohar

                                          1,520316




                                          1,520316












                                          • $begingroup$
                                            This is a hint towards what?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 17:41










                                          • $begingroup$
                                            @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:05










                                          • $begingroup$
                                            I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:08












                                          • $begingroup$
                                            I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:14










                                          • $begingroup$
                                            Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:16




















                                          • $begingroup$
                                            This is a hint towards what?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 17:41










                                          • $begingroup$
                                            @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:05










                                          • $begingroup$
                                            I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:08












                                          • $begingroup$
                                            I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                            $endgroup$
                                            – Mehrdad Zandigohar
                                            Feb 25 '18 at 18:14










                                          • $begingroup$
                                            Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                            $endgroup$
                                            – Jack D'Aurizio
                                            Feb 25 '18 at 18:16


















                                          $begingroup$
                                          This is a hint towards what?
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 17:41




                                          $begingroup$
                                          This is a hint towards what?
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 17:41












                                          $begingroup$
                                          @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                          $endgroup$
                                          – Mehrdad Zandigohar
                                          Feb 25 '18 at 18:05




                                          $begingroup$
                                          @JackD'Aurizio Toward looking differently at how he could solve his problem because this makes solving the problem easier I believe.
                                          $endgroup$
                                          – Mehrdad Zandigohar
                                          Feb 25 '18 at 18:05












                                          $begingroup$
                                          I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 18:08






                                          $begingroup$
                                          I believe that is not true. Both the integrals in the RHS are related to the dilogarithm function evaluated at the third roots of unity. $logfrac{a}{b}=log a-log b$ (for $a,b>0$) is obvious and not really helpful in reaching a "closed" form.
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 18:08














                                          $begingroup$
                                          I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                          $endgroup$
                                          – Mehrdad Zandigohar
                                          Feb 25 '18 at 18:14




                                          $begingroup$
                                          I myself knew how to integrate $ln(sin(x))$ and this made me think this is a better way. It may be obvious but one may make the problem solving easier by this. As omegadot did in his answer too. Anyway, aren't we free to state the things we believe are helpful?
                                          $endgroup$
                                          – Mehrdad Zandigohar
                                          Feb 25 '18 at 18:14












                                          $begingroup$
                                          Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 18:16






                                          $begingroup$
                                          Better to be sure they are: comments are best suited for uncertainty. What is $int_{0}^{pi/5}logsin(x),dx$, for instance?
                                          $endgroup$
                                          – Jack D'Aurizio
                                          Feb 25 '18 at 18:16




















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