Improper integral involving exponential function

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












How can one compute $displaystyle int_0^infty dfracx^3;dxe^x-1$. I tried contour integration replacing $x$ with $z$ but confused about the proper contour for integration.




contour-integration




share|cite|improve this question























    up vote
    0
    down vote

    favorite












    How can one compute $displaystyle int_0^infty dfracx^3;dxe^x-1$. I tried contour integration replacing $x$ with $z$ but confused about the proper contour for integration.




    contour-integration




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      How can one compute $displaystyle int_0^infty dfracx^3;dxe^x-1$. I tried contour integration replacing $x$ with $z$ but confused about the proper contour for integration.




      contour-integration




      share|cite|improve this question











      How can one compute $displaystyle int_0^infty dfracx^3;dxe^x-1$. I tried contour integration replacing $x$ with $z$ but confused about the proper contour for integration.





      contour-integration





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 22 at 5:32









      Purushothaman

      1906




      1906




















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          1
          down vote













          $$int_0^inftyfracx^te^x-1,dx
          =int_0^inftysum_n=1^infty x^te^-nx,dx
          =sum_n=1^inftyint_0^infty x^te^-nx,dx
          =sum_n=1^inftyfracGamma(t+1)n^t+1=Gamma(t+1)zeta(t+1).
          $$






          share|cite|improve this answer




























            up vote
            0
            down vote













            $$int_0^inftyfracx^3e^x-1dx
            =int_0^inftyfracx^3e^-x1-e^-xdx$$
            $frac11-e^-x= sum_n=0^infty e^-nx$
            $$int_0^inftyfracx^3e^x-1dx
            =sum_n=0^inftyint_0^infty x^3e^-(n+1)xdx$$
            $int_0^infty x^3e^-(n+1)xdx=frac6(n+1)^4$
            $$int_0^inftyfracx^3e^x-1dx
            =6sum_n=0^inftyfrac1(n+1)^4=6sum_n=1^inftyfrac1n^4=6zeta(4)=6fracpi^490$$
            $$int_0^inftyfracx^3e^x-1dx
            =fracpi^415$$






            share|cite|improve this answer





















              Your Answer




              StackExchange.ifUsing("editor", function ()
              return StackExchange.using("mathjaxEditing", function ()
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              );
              );
              , "mathjax-editing");

              StackExchange.ready(function()
              var channelOptions =
              tags: "".split(" "),
              id: "69"
              ;
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function()
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled)
              StackExchange.using("snippets", function()
              createEditor();
              );

              else
              createEditor();

              );

              function createEditor()
              StackExchange.prepareEditor(
              heartbeatType: 'answer',
              convertImagesToLinks: true,
              noModals: false,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              );



              );








               

              draft saved


              draft discarded


















              StackExchange.ready(
              function ()
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2859123%2fimproper-integral-involving-exponential-function%23new-answer', 'question_page');

              );

              Post as a guest

















              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              1
              down vote













              $$int_0^inftyfracx^te^x-1,dx
              =int_0^inftysum_n=1^infty x^te^-nx,dx
              =sum_n=1^inftyint_0^infty x^te^-nx,dx
              =sum_n=1^inftyfracGamma(t+1)n^t+1=Gamma(t+1)zeta(t+1).
              $$






              share|cite|improve this answer

























                up vote
                1
                down vote













                $$int_0^inftyfracx^te^x-1,dx
                =int_0^inftysum_n=1^infty x^te^-nx,dx
                =sum_n=1^inftyint_0^infty x^te^-nx,dx
                =sum_n=1^inftyfracGamma(t+1)n^t+1=Gamma(t+1)zeta(t+1).
                $$






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  $$int_0^inftyfracx^te^x-1,dx
                  =int_0^inftysum_n=1^infty x^te^-nx,dx
                  =sum_n=1^inftyint_0^infty x^te^-nx,dx
                  =sum_n=1^inftyfracGamma(t+1)n^t+1=Gamma(t+1)zeta(t+1).
                  $$






                  share|cite|improve this answer













                  $$int_0^inftyfracx^te^x-1,dx
                  =int_0^inftysum_n=1^infty x^te^-nx,dx
                  =sum_n=1^inftyint_0^infty x^te^-nx,dx
                  =sum_n=1^inftyfracGamma(t+1)n^t+1=Gamma(t+1)zeta(t+1).
                  $$







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Jul 22 at 6:20









                  Lord Shark the Unknown

                  85.2k950111




                  85.2k950111




















                      up vote
                      0
                      down vote













                      $$int_0^inftyfracx^3e^x-1dx
                      =int_0^inftyfracx^3e^-x1-e^-xdx$$
                      $frac11-e^-x= sum_n=0^infty e^-nx$
                      $$int_0^inftyfracx^3e^x-1dx
                      =sum_n=0^inftyint_0^infty x^3e^-(n+1)xdx$$
                      $int_0^infty x^3e^-(n+1)xdx=frac6(n+1)^4$
                      $$int_0^inftyfracx^3e^x-1dx
                      =6sum_n=0^inftyfrac1(n+1)^4=6sum_n=1^inftyfrac1n^4=6zeta(4)=6fracpi^490$$
                      $$int_0^inftyfracx^3e^x-1dx
                      =fracpi^415$$






                      share|cite|improve this answer

























                        up vote
                        0
                        down vote













                        $$int_0^inftyfracx^3e^x-1dx
                        =int_0^inftyfracx^3e^-x1-e^-xdx$$
                        $frac11-e^-x= sum_n=0^infty e^-nx$
                        $$int_0^inftyfracx^3e^x-1dx
                        =sum_n=0^inftyint_0^infty x^3e^-(n+1)xdx$$
                        $int_0^infty x^3e^-(n+1)xdx=frac6(n+1)^4$
                        $$int_0^inftyfracx^3e^x-1dx
                        =6sum_n=0^inftyfrac1(n+1)^4=6sum_n=1^inftyfrac1n^4=6zeta(4)=6fracpi^490$$
                        $$int_0^inftyfracx^3e^x-1dx
                        =fracpi^415$$






                        share|cite|improve this answer























                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          $$int_0^inftyfracx^3e^x-1dx
                          =int_0^inftyfracx^3e^-x1-e^-xdx$$
                          $frac11-e^-x= sum_n=0^infty e^-nx$
                          $$int_0^inftyfracx^3e^x-1dx
                          =sum_n=0^inftyint_0^infty x^3e^-(n+1)xdx$$
                          $int_0^infty x^3e^-(n+1)xdx=frac6(n+1)^4$
                          $$int_0^inftyfracx^3e^x-1dx
                          =6sum_n=0^inftyfrac1(n+1)^4=6sum_n=1^inftyfrac1n^4=6zeta(4)=6fracpi^490$$
                          $$int_0^inftyfracx^3e^x-1dx
                          =fracpi^415$$






                          share|cite|improve this answer













                          $$int_0^inftyfracx^3e^x-1dx
                          =int_0^inftyfracx^3e^-x1-e^-xdx$$
                          $frac11-e^-x= sum_n=0^infty e^-nx$
                          $$int_0^inftyfracx^3e^x-1dx
                          =sum_n=0^inftyint_0^infty x^3e^-(n+1)xdx$$
                          $int_0^infty x^3e^-(n+1)xdx=frac6(n+1)^4$
                          $$int_0^inftyfracx^3e^x-1dx
                          =6sum_n=0^inftyfrac1(n+1)^4=6sum_n=1^inftyfrac1n^4=6zeta(4)=6fracpi^490$$
                          $$int_0^inftyfracx^3e^x-1dx
                          =fracpi^415$$







                          share|cite|improve this answer













                          share|cite|improve this answer



                          share|cite|improve this answer











                          answered Jul 22 at 7:19









                          JJacquelin

                          39.9k21649




                          39.9k21649






















                               

                              draft saved


                              draft discarded


























                               


                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function ()
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2859123%2fimproper-integral-involving-exponential-function%23new-answer', 'question_page');

                              );

                              Post as a guest
































































                              Comments

                              Post a Comment

                              Popular posts from this blog

                              What is the equation of a 3D cone with generalised tilt?

                              Image
                              Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                              Read more

                              Color the edges and diagonals of a regular polygon

                              Image
                              Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                              Read more

                              Relationship between determinant of matrix and determinant of adjoint?

                              Image
                              Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                              Read more