Proving $int x^x dx$ is not elementary

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Please verify if my reasoning is correct.



Write $x^x$ as $e^xln x$. Then let $f(x)=1$ and $g(x)=xln x$.



$R'(x) + g'(x)R(x) = f(x)$ will be $R'(x) + (ln x+1)R(x) = 1$, which solution is $R(x) = c_1x^-x + x^-x int_1^x u^u du$, then there's no rational solution $R$ to the ode, then $int x^x dx$ is not elementary, by Liouville's Theorem Corollary (page 24).



Thanks!




calculus integration indefinite-integrals elementary-functions




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    up vote
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    favorite
    1












    Please verify if my reasoning is correct.



    Write $x^x$ as $e^xln x$. Then let $f(x)=1$ and $g(x)=xln x$.



    $R'(x) + g'(x)R(x) = f(x)$ will be $R'(x) + (ln x+1)R(x) = 1$, which solution is $R(x) = c_1x^-x + x^-x int_1^x u^u du$, then there's no rational solution $R$ to the ode, then $int x^x dx$ is not elementary, by Liouville's Theorem Corollary (page 24).



    Thanks!




    calculus integration indefinite-integrals elementary-functions




    share|cite|improve this question





















      up vote
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      down vote

      favorite
      1









      up vote
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      down vote

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      Please verify if my reasoning is correct.



      Write $x^x$ as $e^xln x$. Then let $f(x)=1$ and $g(x)=xln x$.



      $R'(x) + g'(x)R(x) = f(x)$ will be $R'(x) + (ln x+1)R(x) = 1$, which solution is $R(x) = c_1x^-x + x^-x int_1^x u^u du$, then there's no rational solution $R$ to the ode, then $int x^x dx$ is not elementary, by Liouville's Theorem Corollary (page 24).



      Thanks!




      calculus integration indefinite-integrals elementary-functions




      share|cite|improve this question











      Please verify if my reasoning is correct.



      Write $x^x$ as $e^xln x$. Then let $f(x)=1$ and $g(x)=xln x$.



      $R'(x) + g'(x)R(x) = f(x)$ will be $R'(x) + (ln x+1)R(x) = 1$, which solution is $R(x) = c_1x^-x + x^-x int_1^x u^u du$, then there's no rational solution $R$ to the ode, then $int x^x dx$ is not elementary, by Liouville's Theorem Corollary (page 24).



      Thanks!





      calculus integration indefinite-integrals elementary-functions





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      asked Aug 6 at 4:21









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