test the integral $int_{0}^{infty} frac {x}{3x^4 + 5x^2 +1}dx$ for convergence












0












$begingroup$


test the integral $int_{0}^{infty} frac {x}{3x^4 + 5x^2 +1}dx$ for convergence.



My thought



Can I compare it with 1/(3x^4)?



Any hints for the solution are appreciated!






real-analysis calculus convergence improper-integrals







share|cite|improve this question









$endgroup$












  • $begingroup$
    Is $3x^4+5x^2+1>x^4$?
    $endgroup$
    – John Wayland Bales
    Dec 7 '18 at 6:16






  • 2




    $begingroup$
    you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
    $endgroup$
    – Lau
    Dec 7 '18 at 6:19












  • $begingroup$
    @Lau but I have to prove this fact first ....... is it proved here on math stack ?
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:40






  • 1




    $begingroup$
    This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
    $endgroup$
    – Shubham Johri
    Dec 7 '18 at 6:44












  • $begingroup$
    I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:55
















0












$begingroup$


test the integral $int_{0}^{infty} frac {x}{3x^4 + 5x^2 +1}dx$ for convergence.



My thought



Can I compare it with 1/(3x^4)?



Any hints for the solution are appreciated!






real-analysis calculus convergence improper-integrals







share|cite|improve this question









$endgroup$












  • $begingroup$
    Is $3x^4+5x^2+1>x^4$?
    $endgroup$
    – John Wayland Bales
    Dec 7 '18 at 6:16






  • 2




    $begingroup$
    you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
    $endgroup$
    – Lau
    Dec 7 '18 at 6:19












  • $begingroup$
    @Lau but I have to prove this fact first ....... is it proved here on math stack ?
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:40






  • 1




    $begingroup$
    This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
    $endgroup$
    – Shubham Johri
    Dec 7 '18 at 6:44












  • $begingroup$
    I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:55














0












0








0





$begingroup$


test the integral $int_{0}^{infty} frac {x}{3x^4 + 5x^2 +1}dx$ for convergence.



My thought



Can I compare it with 1/(3x^4)?



Any hints for the solution are appreciated!






real-analysis calculus convergence improper-integrals







share|cite|improve this question









$endgroup$




test the integral $int_{0}^{infty} frac {x}{3x^4 + 5x^2 +1}dx$ for convergence.



My thought



Can I compare it with 1/(3x^4)?



Any hints for the solution are appreciated!






real-analysis calculus convergence improper-integrals




real-analysis calculus convergence improper-integrals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 7 '18 at 6:10









hopefullyhopefully

310113




310113












  • $begingroup$
    Is $3x^4+5x^2+1>x^4$?
    $endgroup$
    – John Wayland Bales
    Dec 7 '18 at 6:16






  • 2




    $begingroup$
    you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
    $endgroup$
    – Lau
    Dec 7 '18 at 6:19












  • $begingroup$
    @Lau but I have to prove this fact first ....... is it proved here on math stack ?
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:40






  • 1




    $begingroup$
    This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
    $endgroup$
    – Shubham Johri
    Dec 7 '18 at 6:44












  • $begingroup$
    I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:55


















  • $begingroup$
    Is $3x^4+5x^2+1>x^4$?
    $endgroup$
    – John Wayland Bales
    Dec 7 '18 at 6:16






  • 2




    $begingroup$
    you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
    $endgroup$
    – Lau
    Dec 7 '18 at 6:19












  • $begingroup$
    @Lau but I have to prove this fact first ....... is it proved here on math stack ?
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:40






  • 1




    $begingroup$
    This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
    $endgroup$
    – Shubham Johri
    Dec 7 '18 at 6:44












  • $begingroup$
    I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
    $endgroup$
    – hopefully
    Dec 7 '18 at 6:55
















$begingroup$
Is $3x^4+5x^2+1>x^4$?
$endgroup$
– John Wayland Bales
Dec 7 '18 at 6:16




$begingroup$
Is $3x^4+5x^2+1>x^4$?
$endgroup$
– John Wayland Bales
Dec 7 '18 at 6:16




2




2




$begingroup$
you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
$endgroup$
– Lau
Dec 7 '18 at 6:19






$begingroup$
you can try $$frac{x}{3x^4+5x^2+1}leq frac{1}{x^3},,,,xgeq 0$$
$endgroup$
– Lau
Dec 7 '18 at 6:19














$begingroup$
@Lau but I have to prove this fact first ....... is it proved here on math stack ?
$endgroup$
– hopefully
Dec 7 '18 at 6:40




$begingroup$
@Lau but I have to prove this fact first ....... is it proved here on math stack ?
$endgroup$
– hopefully
Dec 7 '18 at 6:40




1




1




$begingroup$
This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
$endgroup$
– Shubham Johri
Dec 7 '18 at 6:44






$begingroup$
This can be easily seen:$forall xgeq0, frac x{3x^4+5x^2+1}leqfrac x{3x^4}leqfrac x{x^4} because$ the terms we ignore in the denominator are all positive. By ignoring them, we make the denominator smaller and thus the value of the ratio larger.
$endgroup$
– Shubham Johri
Dec 7 '18 at 6:44














$begingroup$
I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
$endgroup$
– hopefully
Dec 7 '18 at 6:55




$begingroup$
I am not speaking about this @Lau I am speaking about proving the convergence of an integral that is different from the required one (the initial point differes)
$endgroup$
– hopefully
Dec 7 '18 at 6:55










1 Answer
1






active

oldest

votes


















2












$begingroup$

$I=int_0^infty frac x{3x^4 + 5x^2 +1}dx$



$forall x>0, 0leqfrac x{3x^4 + 5x^2 +1}leqfrac x{x^4}=frac1{x^3}$



$implies 0leq I=int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac x{3x^4 + 5x^2 +1}dx\ leqint_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+frac12$



Since $frac x{3x^4 + 5x^2 +1}$ is continuous on $[0,1]$ (the denominator has no real zeroes), it is bounded there and so is the proper integral $int_0^1 frac x{3x^4 + 5x^2 +1}dx$.






share|cite|improve this answer











$endgroup$













    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',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    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%2f3029548%2ftest-the-integral-int-0-infty-frac-x3x4-5x2-1dx-for-convergenc%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    $I=int_0^infty frac x{3x^4 + 5x^2 +1}dx$



    $forall x>0, 0leqfrac x{3x^4 + 5x^2 +1}leqfrac x{x^4}=frac1{x^3}$



    $implies 0leq I=int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac x{3x^4 + 5x^2 +1}dx\ leqint_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+frac12$



    Since $frac x{3x^4 + 5x^2 +1}$ is continuous on $[0,1]$ (the denominator has no real zeroes), it is bounded there and so is the proper integral $int_0^1 frac x{3x^4 + 5x^2 +1}dx$.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      $I=int_0^infty frac x{3x^4 + 5x^2 +1}dx$



      $forall x>0, 0leqfrac x{3x^4 + 5x^2 +1}leqfrac x{x^4}=frac1{x^3}$



      $implies 0leq I=int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac x{3x^4 + 5x^2 +1}dx\ leqint_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+frac12$



      Since $frac x{3x^4 + 5x^2 +1}$ is continuous on $[0,1]$ (the denominator has no real zeroes), it is bounded there and so is the proper integral $int_0^1 frac x{3x^4 + 5x^2 +1}dx$.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        $I=int_0^infty frac x{3x^4 + 5x^2 +1}dx$



        $forall x>0, 0leqfrac x{3x^4 + 5x^2 +1}leqfrac x{x^4}=frac1{x^3}$



        $implies 0leq I=int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac x{3x^4 + 5x^2 +1}dx\ leqint_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+frac12$



        Since $frac x{3x^4 + 5x^2 +1}$ is continuous on $[0,1]$ (the denominator has no real zeroes), it is bounded there and so is the proper integral $int_0^1 frac x{3x^4 + 5x^2 +1}dx$.






        share|cite|improve this answer











        $endgroup$



        $I=int_0^infty frac x{3x^4 + 5x^2 +1}dx$



        $forall x>0, 0leqfrac x{3x^4 + 5x^2 +1}leqfrac x{x^4}=frac1{x^3}$



        $implies 0leq I=int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac x{3x^4 + 5x^2 +1}dx\ leqint_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+int_1^infty frac 1{x^3}dx\ =int_0^1 frac x{3x^4 + 5x^2 +1}dx+frac12$



        Since $frac x{3x^4 + 5x^2 +1}$ is continuous on $[0,1]$ (the denominator has no real zeroes), it is bounded there and so is the proper integral $int_0^1 frac x{3x^4 + 5x^2 +1}dx$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 7 '18 at 6:46

























        answered Dec 7 '18 at 6:39









        Shubham JohriShubham Johri

        5,057717




        5,057717






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029548%2ftest-the-integral-int-0-infty-frac-x3x4-5x2-1dx-for-convergenc%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Probability when a professor distributes a quiz and homework assignment to a class of n students.

            Image
            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
            Read more

            Aardman Animations

            Image
            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
            Read more

            Are they similar matrix

            Image
            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
            Read more