Calculating the integral $int_{-infty}^{infty} xsin(x)/(x^2+1)^2 dx$












1












$begingroup$


I need help calculation the integral $int_{-infty}^{infty} xsin(x)/(x^2+1)^2 dx$



I am getting confused on how to integrate around $pm i$. I think I'll have to use $gamma = Re^{itheta}$ where $|R|leq n$ for $ngeq1$










share|cite|improve this question











$endgroup$












  • $begingroup$
    Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
    $endgroup$
    – Mark
    Jan 4 at 16:22












  • $begingroup$
    You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
    $endgroup$
    – Ricardo770
    Jan 4 at 16:50












  • $begingroup$
    The result should be $$frac{pi}{2e}$$
    $endgroup$
    – Dr. Sonnhard Graubner
    Jan 4 at 17:12
















1












$begingroup$


I need help calculation the integral $int_{-infty}^{infty} xsin(x)/(x^2+1)^2 dx$



I am getting confused on how to integrate around $pm i$. I think I'll have to use $gamma = Re^{itheta}$ where $|R|leq n$ for $ngeq1$










share|cite|improve this question











$endgroup$












  • $begingroup$
    Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
    $endgroup$
    – Mark
    Jan 4 at 16:22












  • $begingroup$
    You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
    $endgroup$
    – Ricardo770
    Jan 4 at 16:50












  • $begingroup$
    The result should be $$frac{pi}{2e}$$
    $endgroup$
    – Dr. Sonnhard Graubner
    Jan 4 at 17:12














1












1








1





$begingroup$


I need help calculation the integral $int_{-infty}^{infty} xsin(x)/(x^2+1)^2 dx$



I am getting confused on how to integrate around $pm i$. I think I'll have to use $gamma = Re^{itheta}$ where $|R|leq n$ for $ngeq1$










share|cite|improve this question











$endgroup$




I need help calculation the integral $int_{-infty}^{infty} xsin(x)/(x^2+1)^2 dx$



I am getting confused on how to integrate around $pm i$. I think I'll have to use $gamma = Re^{itheta}$ where $|R|leq n$ for $ngeq1$







real-analysis integration complex-analysis analysis






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edited Jan 4 at 17:49









Bernard

123k741116




123k741116










asked Jan 4 at 16:18









user628226user628226

575




575












  • $begingroup$
    Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
    $endgroup$
    – Mark
    Jan 4 at 16:22












  • $begingroup$
    You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
    $endgroup$
    – Ricardo770
    Jan 4 at 16:50












  • $begingroup$
    The result should be $$frac{pi}{2e}$$
    $endgroup$
    – Dr. Sonnhard Graubner
    Jan 4 at 17:12


















  • $begingroup$
    Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
    $endgroup$
    – Mark
    Jan 4 at 16:22












  • $begingroup$
    You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
    $endgroup$
    – Ricardo770
    Jan 4 at 16:50












  • $begingroup$
    The result should be $$frac{pi}{2e}$$
    $endgroup$
    – Dr. Sonnhard Graubner
    Jan 4 at 17:12
















$begingroup$
Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
$endgroup$
– Mark
Jan 4 at 16:22






$begingroup$
Yes, half a circle is a good idea. Prove that when $Rtoinfty$ the integral goes to zero. You don't need anything around $-i$, that will only be more work.
$endgroup$
– Mark
Jan 4 at 16:22














$begingroup$
You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
$endgroup$
– Ricardo770
Jan 4 at 16:50






$begingroup$
You could try solving $int_{-infty}^{infty} frac{xsin(alpha{x})}{(x^{2}+1)^{2}}dx=-frac{d}{dalpha}int_{-infty}^{infty} frac{cos(alpha{x})}{(x^{2}+1)^{2}}dx$ and then let $alpharightarrow1$. It might be easier.
$endgroup$
– Ricardo770
Jan 4 at 16:50














$begingroup$
The result should be $$frac{pi}{2e}$$
$endgroup$
– Dr. Sonnhard Graubner
Jan 4 at 17:12




$begingroup$
The result should be $$frac{pi}{2e}$$
$endgroup$
– Dr. Sonnhard Graubner
Jan 4 at 17:12










2 Answers
2






active

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2












$begingroup$

$$int_{-infty}^inftyfrac{xsin x}{(x^2+1)^2}mathrm dx=mathfrak Ileft{int_{-infty}^inftyfrac{xe^{ix}}{(x^2+1)^2}mathrm dxright}$$



Let us compute this integral. It is given by a contour in the complex plane that runs along the real axis. Since the exponential on top decays as $|x|toinfty$ in the upper half plane, this is where we choose to close the contour. Therefore we have a semicircular contour in the upper half plane, of the form $Re^{itheta}$ for $R>1$ and $thetain[0,pi]$. Let us denote this contour by $gamma_R$.



$$lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dx=int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx+lim_{Rtoinfty}int_0^pifrac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetatag1$$



Now we want to show that this second integral is zero in the limit as $Rtoinfty$.



$$begin{align}left|int_0^pi frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetaright|&leint_0^pileft|frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}right|mathrm dtheta\&leint_0^pileft|frac{R^2e^{-Rsin theta}}{(R^2-1)^2}right|mathrm dthetaend{align}$$



$int_0^pi e^{-Rsintheta}mathrm dtheta$ is finite, and in the limit as $Rtoinfty$, the prefactor tends to zero, so the second integral in $(1)$ is zero.



So we have $$int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx=lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dxtag2$$



We evaluate the RHS of $(2)$ by residue calculus. The contour encloses a pole at $x=i$, of order $2$. We evaluate the residue at the pole.



$$text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}(x-i)^2}{(x^2+1)^2}right)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}}{(x+i)^2}right)=frac1{4e}$$after some calculations.



Therefore the RHS of $(2)$ is $$2pi icdot text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=frac{pi i }{2e}$$



Since the integral we wish to calculate is the imaginary part of the LHS of $(2)$, the final answer is $$frac{pi}{2e}.$$






share|cite|improve this answer









$endgroup$





















    3












    $begingroup$

    Let us define
    $$ g(a) = int_{0}^{+infty}frac{xsin(ax)}{(x^2+1)^2},dx $$
    for $ainmathbb{R}^+$. Our integral is clearly $2,g(1)$ by parity. We have
    $$ mathcal L g(s) = int_{0}^{+infty}frac{x^2}{(s^2+x^2)(1+x^2)^2},dx = frac{pi}{4(1+s)^2} $$
    by partial fraction decomposition, and
    $$ g(a) = frac{pi a}{4}e^{-a} $$
    by the inverse Laplace transform. It follows that the original integral equals $color{blue}{frac{pi}{2e}}$.






    share|cite|improve this answer









    $endgroup$














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      2 Answers
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      2 Answers
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      $begingroup$

      $$int_{-infty}^inftyfrac{xsin x}{(x^2+1)^2}mathrm dx=mathfrak Ileft{int_{-infty}^inftyfrac{xe^{ix}}{(x^2+1)^2}mathrm dxright}$$



      Let us compute this integral. It is given by a contour in the complex plane that runs along the real axis. Since the exponential on top decays as $|x|toinfty$ in the upper half plane, this is where we choose to close the contour. Therefore we have a semicircular contour in the upper half plane, of the form $Re^{itheta}$ for $R>1$ and $thetain[0,pi]$. Let us denote this contour by $gamma_R$.



      $$lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dx=int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx+lim_{Rtoinfty}int_0^pifrac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetatag1$$



      Now we want to show that this second integral is zero in the limit as $Rtoinfty$.



      $$begin{align}left|int_0^pi frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetaright|&leint_0^pileft|frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}right|mathrm dtheta\&leint_0^pileft|frac{R^2e^{-Rsin theta}}{(R^2-1)^2}right|mathrm dthetaend{align}$$



      $int_0^pi e^{-Rsintheta}mathrm dtheta$ is finite, and in the limit as $Rtoinfty$, the prefactor tends to zero, so the second integral in $(1)$ is zero.



      So we have $$int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx=lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dxtag2$$



      We evaluate the RHS of $(2)$ by residue calculus. The contour encloses a pole at $x=i$, of order $2$. We evaluate the residue at the pole.



      $$text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}(x-i)^2}{(x^2+1)^2}right)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}}{(x+i)^2}right)=frac1{4e}$$after some calculations.



      Therefore the RHS of $(2)$ is $$2pi icdot text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=frac{pi i }{2e}$$



      Since the integral we wish to calculate is the imaginary part of the LHS of $(2)$, the final answer is $$frac{pi}{2e}.$$






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        $$int_{-infty}^inftyfrac{xsin x}{(x^2+1)^2}mathrm dx=mathfrak Ileft{int_{-infty}^inftyfrac{xe^{ix}}{(x^2+1)^2}mathrm dxright}$$



        Let us compute this integral. It is given by a contour in the complex plane that runs along the real axis. Since the exponential on top decays as $|x|toinfty$ in the upper half plane, this is where we choose to close the contour. Therefore we have a semicircular contour in the upper half plane, of the form $Re^{itheta}$ for $R>1$ and $thetain[0,pi]$. Let us denote this contour by $gamma_R$.



        $$lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dx=int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx+lim_{Rtoinfty}int_0^pifrac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetatag1$$



        Now we want to show that this second integral is zero in the limit as $Rtoinfty$.



        $$begin{align}left|int_0^pi frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetaright|&leint_0^pileft|frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}right|mathrm dtheta\&leint_0^pileft|frac{R^2e^{-Rsin theta}}{(R^2-1)^2}right|mathrm dthetaend{align}$$



        $int_0^pi e^{-Rsintheta}mathrm dtheta$ is finite, and in the limit as $Rtoinfty$, the prefactor tends to zero, so the second integral in $(1)$ is zero.



        So we have $$int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx=lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dxtag2$$



        We evaluate the RHS of $(2)$ by residue calculus. The contour encloses a pole at $x=i$, of order $2$. We evaluate the residue at the pole.



        $$text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}(x-i)^2}{(x^2+1)^2}right)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}}{(x+i)^2}right)=frac1{4e}$$after some calculations.



        Therefore the RHS of $(2)$ is $$2pi icdot text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=frac{pi i }{2e}$$



        Since the integral we wish to calculate is the imaginary part of the LHS of $(2)$, the final answer is $$frac{pi}{2e}.$$






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          $$int_{-infty}^inftyfrac{xsin x}{(x^2+1)^2}mathrm dx=mathfrak Ileft{int_{-infty}^inftyfrac{xe^{ix}}{(x^2+1)^2}mathrm dxright}$$



          Let us compute this integral. It is given by a contour in the complex plane that runs along the real axis. Since the exponential on top decays as $|x|toinfty$ in the upper half plane, this is where we choose to close the contour. Therefore we have a semicircular contour in the upper half plane, of the form $Re^{itheta}$ for $R>1$ and $thetain[0,pi]$. Let us denote this contour by $gamma_R$.



          $$lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dx=int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx+lim_{Rtoinfty}int_0^pifrac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetatag1$$



          Now we want to show that this second integral is zero in the limit as $Rtoinfty$.



          $$begin{align}left|int_0^pi frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetaright|&leint_0^pileft|frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}right|mathrm dtheta\&leint_0^pileft|frac{R^2e^{-Rsin theta}}{(R^2-1)^2}right|mathrm dthetaend{align}$$



          $int_0^pi e^{-Rsintheta}mathrm dtheta$ is finite, and in the limit as $Rtoinfty$, the prefactor tends to zero, so the second integral in $(1)$ is zero.



          So we have $$int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx=lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dxtag2$$



          We evaluate the RHS of $(2)$ by residue calculus. The contour encloses a pole at $x=i$, of order $2$. We evaluate the residue at the pole.



          $$text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}(x-i)^2}{(x^2+1)^2}right)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}}{(x+i)^2}right)=frac1{4e}$$after some calculations.



          Therefore the RHS of $(2)$ is $$2pi icdot text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=frac{pi i }{2e}$$



          Since the integral we wish to calculate is the imaginary part of the LHS of $(2)$, the final answer is $$frac{pi}{2e}.$$






          share|cite|improve this answer









          $endgroup$



          $$int_{-infty}^inftyfrac{xsin x}{(x^2+1)^2}mathrm dx=mathfrak Ileft{int_{-infty}^inftyfrac{xe^{ix}}{(x^2+1)^2}mathrm dxright}$$



          Let us compute this integral. It is given by a contour in the complex plane that runs along the real axis. Since the exponential on top decays as $|x|toinfty$ in the upper half plane, this is where we choose to close the contour. Therefore we have a semicircular contour in the upper half plane, of the form $Re^{itheta}$ for $R>1$ and $thetain[0,pi]$. Let us denote this contour by $gamma_R$.



          $$lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dx=int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx+lim_{Rtoinfty}int_0^pifrac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetatag1$$



          Now we want to show that this second integral is zero in the limit as $Rtoinfty$.



          $$begin{align}left|int_0^pi frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}mathrm dthetaright|&leint_0^pileft|frac{Re^{itheta}e^{iRe^{itheta}}}{(R^2e^{2itheta}+1)^2}iRe^{itheta}right|mathrm dtheta\&leint_0^pileft|frac{R^2e^{-Rsin theta}}{(R^2-1)^2}right|mathrm dthetaend{align}$$



          $int_0^pi e^{-Rsintheta}mathrm dtheta$ is finite, and in the limit as $Rtoinfty$, the prefactor tends to zero, so the second integral in $(1)$ is zero.



          So we have $$int_{-infty}^{infty}frac{xe^{ix}}{(x^2+1)^2}mathrm dx=lim_{Rtoinfty}int_{gamma_R} frac{xe^{ix}}{(x^2+1)^2}mathrm dxtag2$$



          We evaluate the RHS of $(2)$ by residue calculus. The contour encloses a pole at $x=i$, of order $2$. We evaluate the residue at the pole.



          $$text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}(x-i)^2}{(x^2+1)^2}right)=lim_{xto i}frac{d}{dx}left(frac{xe^{ix}}{(x+i)^2}right)=frac1{4e}$$after some calculations.



          Therefore the RHS of $(2)$ is $$2pi icdot text{Res}left(frac{xe^{ix}}{(x^2+1)^2},iright)=frac{pi i }{2e}$$



          Since the integral we wish to calculate is the imaginary part of the LHS of $(2)$, the final answer is $$frac{pi}{2e}.$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 4 at 17:22









          John DoeJohn Doe

          11.7k11239




          11.7k11239























              3












              $begingroup$

              Let us define
              $$ g(a) = int_{0}^{+infty}frac{xsin(ax)}{(x^2+1)^2},dx $$
              for $ainmathbb{R}^+$. Our integral is clearly $2,g(1)$ by parity. We have
              $$ mathcal L g(s) = int_{0}^{+infty}frac{x^2}{(s^2+x^2)(1+x^2)^2},dx = frac{pi}{4(1+s)^2} $$
              by partial fraction decomposition, and
              $$ g(a) = frac{pi a}{4}e^{-a} $$
              by the inverse Laplace transform. It follows that the original integral equals $color{blue}{frac{pi}{2e}}$.






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                Let us define
                $$ g(a) = int_{0}^{+infty}frac{xsin(ax)}{(x^2+1)^2},dx $$
                for $ainmathbb{R}^+$. Our integral is clearly $2,g(1)$ by parity. We have
                $$ mathcal L g(s) = int_{0}^{+infty}frac{x^2}{(s^2+x^2)(1+x^2)^2},dx = frac{pi}{4(1+s)^2} $$
                by partial fraction decomposition, and
                $$ g(a) = frac{pi a}{4}e^{-a} $$
                by the inverse Laplace transform. It follows that the original integral equals $color{blue}{frac{pi}{2e}}$.






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Let us define
                  $$ g(a) = int_{0}^{+infty}frac{xsin(ax)}{(x^2+1)^2},dx $$
                  for $ainmathbb{R}^+$. Our integral is clearly $2,g(1)$ by parity. We have
                  $$ mathcal L g(s) = int_{0}^{+infty}frac{x^2}{(s^2+x^2)(1+x^2)^2},dx = frac{pi}{4(1+s)^2} $$
                  by partial fraction decomposition, and
                  $$ g(a) = frac{pi a}{4}e^{-a} $$
                  by the inverse Laplace transform. It follows that the original integral equals $color{blue}{frac{pi}{2e}}$.






                  share|cite|improve this answer









                  $endgroup$



                  Let us define
                  $$ g(a) = int_{0}^{+infty}frac{xsin(ax)}{(x^2+1)^2},dx $$
                  for $ainmathbb{R}^+$. Our integral is clearly $2,g(1)$ by parity. We have
                  $$ mathcal L g(s) = int_{0}^{+infty}frac{x^2}{(s^2+x^2)(1+x^2)^2},dx = frac{pi}{4(1+s)^2} $$
                  by partial fraction decomposition, and
                  $$ g(a) = frac{pi a}{4}e^{-a} $$
                  by the inverse Laplace transform. It follows that the original integral equals $color{blue}{frac{pi}{2e}}$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 4 at 18:06









                  Jack D'AurizioJack D'Aurizio

                  292k33284672




                  292k33284672






























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