Trig Subsitution When There's No Square RootIntegral of $int sqrt1-4x^2$Struggling with an integral with trig substitutionUsing trig substitution, how do you solve an integral when the leading coefficient under the radical isn't 1?How do you solve for bounds when performing trig substitution and knowing solving the trig functions yields multiple correct values of $theta$?Domain of a square root natural log functionHelp with an integral of binomial differentialTrig substitution for $int fracx^2dxsqrt4 - x^2$Trigonometric integral with square root (residue theorem)Trig Subs issuesNot getting the right answer with alternate completing the square method on $intfracx^2sqrt3+4x-4x^2^3dx$

Why is there an extra space when I type "ls" in the Desktop directory?

What will happen if my luggage gets delayed?

What do *foreign films* mean for an American?

The meaning of ‘otherwise’

Source permutation

For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

What is better: yes / no radio, or simple checkbox?

Is divide-by-zero a security vulnerability?

MySQL importing CSV files really slow

Proving a statement about real numbers

Why aren't there more Gauls like Obelix?

Signed and unsigned numbers

I need help with tikz tree node and label, offsets and inclination

How to resolve: Reviewer #1 says remove section X vs. Reviewer #2 says expand section X

After `ssh` without `-X` to a machine, is it possible to change `$DISPLAY` to make it work like `ssh -X`?

Is it possible that a question has only two answers?

How many characters using PHB rules does it take to be able to have access to any PHB spell at the start of an adventuring day?

Is it possible to find 2014 distinct positive integers whose sum is divisible by each of them?

Does Christianity allow for believing on someone else's behalf?

What ability score modifier does a javelin's damage use?

How can I manipulate the output of Information?

Is it safe to abruptly remove Arduino power?

What is this diamond of every day?

Doubts in understanding some concepts of potential energy



Trig Subsitution When There's No Square Root


Integral of $int sqrt1-4x^2$Struggling with an integral with trig substitutionUsing trig substitution, how do you solve an integral when the leading coefficient under the radical isn't 1?How do you solve for bounds when performing trig substitution and knowing solving the trig functions yields multiple correct values of $theta$?Domain of a square root natural log functionHelp with an integral of binomial differentialTrig substitution for $int fracx^2dxsqrt4 - x^2$Trigonometric integral with square root (residue theorem)Trig Subs issuesNot getting the right answer with alternate completing the square method on $intfracx^2sqrt3+4x-4x^2^3dx$













2












$begingroup$


I would say I'm rather good at doing trig substitution when there is a square root, but when there isn't one, I'm lost.



I'm currently trying to solve the following question:



$Ar int_a^infty fracdx(r^2+x^2)^(3/2)$



Anyway, so far, I have that:



$x = rtan theta$



$dx = rsec^2 theta$



$sqrt (r^2+x^2) = rsectheta$



The triangle I based the above values on:



Triangle I based the above values on



Given that $(r^2+x^2)^(3/2)$ can be rewritten as $ (sqrtr^2+x^2)^3$, I begin to solve.
Please pretend I have $lim limits_b to infty$ in front of every line please.



= $Ar int_a^b fracrsec^2theta(rsectheta)^3dtheta$



= $Ar int_a^b fracrsec^2thetar^3sec^6thetadtheta$



= $fracAr int_a^b frac1sec^4thetadtheta$



= $fracAr int_a^b cos^4theta dtheta$



= $fracAr int_a^b (cos^2theta)^2 dtheta$



= $fracAr int_a^b [ frac121+cos(2theta)) ]^2dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) + cos^2(2theta) dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) dtheta quad+quad fracA4r int_a^b cos^2(2theta) dtheta$



And from there it gets really messed up and I end up with a weird semi-final answer of $fracA4r[2theta+sin(2theta)] + fracA32r [4theta+sin(4theta)]$ which is wrong after I make substitutions.



I already know that the final answer is $fracAr(1-fracasqrtr^2+a^2)$, but I really want to understand this.










share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
    $endgroup$
    – Kay K.
    5 hours ago
















2












$begingroup$


I would say I'm rather good at doing trig substitution when there is a square root, but when there isn't one, I'm lost.



I'm currently trying to solve the following question:



$Ar int_a^infty fracdx(r^2+x^2)^(3/2)$



Anyway, so far, I have that:



$x = rtan theta$



$dx = rsec^2 theta$



$sqrt (r^2+x^2) = rsectheta$



The triangle I based the above values on:



Triangle I based the above values on



Given that $(r^2+x^2)^(3/2)$ can be rewritten as $ (sqrtr^2+x^2)^3$, I begin to solve.
Please pretend I have $lim limits_b to infty$ in front of every line please.



= $Ar int_a^b fracrsec^2theta(rsectheta)^3dtheta$



= $Ar int_a^b fracrsec^2thetar^3sec^6thetadtheta$



= $fracAr int_a^b frac1sec^4thetadtheta$



= $fracAr int_a^b cos^4theta dtheta$



= $fracAr int_a^b (cos^2theta)^2 dtheta$



= $fracAr int_a^b [ frac121+cos(2theta)) ]^2dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) + cos^2(2theta) dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) dtheta quad+quad fracA4r int_a^b cos^2(2theta) dtheta$



And from there it gets really messed up and I end up with a weird semi-final answer of $fracA4r[2theta+sin(2theta)] + fracA32r [4theta+sin(4theta)]$ which is wrong after I make substitutions.



I already know that the final answer is $fracAr(1-fracasqrtr^2+a^2)$, but I really want to understand this.










share|cite|improve this question











$endgroup$







  • 2




    $begingroup$
    The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
    $endgroup$
    – Kay K.
    5 hours ago














2












2








2





$begingroup$


I would say I'm rather good at doing trig substitution when there is a square root, but when there isn't one, I'm lost.



I'm currently trying to solve the following question:



$Ar int_a^infty fracdx(r^2+x^2)^(3/2)$



Anyway, so far, I have that:



$x = rtan theta$



$dx = rsec^2 theta$



$sqrt (r^2+x^2) = rsectheta$



The triangle I based the above values on:



Triangle I based the above values on



Given that $(r^2+x^2)^(3/2)$ can be rewritten as $ (sqrtr^2+x^2)^3$, I begin to solve.
Please pretend I have $lim limits_b to infty$ in front of every line please.



= $Ar int_a^b fracrsec^2theta(rsectheta)^3dtheta$



= $Ar int_a^b fracrsec^2thetar^3sec^6thetadtheta$



= $fracAr int_a^b frac1sec^4thetadtheta$



= $fracAr int_a^b cos^4theta dtheta$



= $fracAr int_a^b (cos^2theta)^2 dtheta$



= $fracAr int_a^b [ frac121+cos(2theta)) ]^2dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) + cos^2(2theta) dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) dtheta quad+quad fracA4r int_a^b cos^2(2theta) dtheta$



And from there it gets really messed up and I end up with a weird semi-final answer of $fracA4r[2theta+sin(2theta)] + fracA32r [4theta+sin(4theta)]$ which is wrong after I make substitutions.



I already know that the final answer is $fracAr(1-fracasqrtr^2+a^2)$, but I really want to understand this.










share|cite|improve this question











$endgroup$




I would say I'm rather good at doing trig substitution when there is a square root, but when there isn't one, I'm lost.



I'm currently trying to solve the following question:



$Ar int_a^infty fracdx(r^2+x^2)^(3/2)$



Anyway, so far, I have that:



$x = rtan theta$



$dx = rsec^2 theta$



$sqrt (r^2+x^2) = rsectheta$



The triangle I based the above values on:



Triangle I based the above values on



Given that $(r^2+x^2)^(3/2)$ can be rewritten as $ (sqrtr^2+x^2)^3$, I begin to solve.
Please pretend I have $lim limits_b to infty$ in front of every line please.



= $Ar int_a^b fracrsec^2theta(rsectheta)^3dtheta$



= $Ar int_a^b fracrsec^2thetar^3sec^6thetadtheta$



= $fracAr int_a^b frac1sec^4thetadtheta$



= $fracAr int_a^b cos^4theta dtheta$



= $fracAr int_a^b (cos^2theta)^2 dtheta$



= $fracAr int_a^b [ frac121+cos(2theta)) ]^2dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) + cos^2(2theta) dtheta$



= $fracA4r int_a^b 1 + 2cos(2theta) dtheta quad+quad fracA4r int_a^b cos^2(2theta) dtheta$



And from there it gets really messed up and I end up with a weird semi-final answer of $fracA4r[2theta+sin(2theta)] + fracA32r [4theta+sin(4theta)]$ which is wrong after I make substitutions.



I already know that the final answer is $fracAr(1-fracasqrtr^2+a^2)$, but I really want to understand this.







calculus integration improper-integrals trigonometric-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 1 min ago









Aaron Hall

711615




711615










asked 6 hours ago









CodingMeeCodingMee

204




204







  • 2




    $begingroup$
    The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
    $endgroup$
    – Kay K.
    5 hours ago













  • 2




    $begingroup$
    The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
    $endgroup$
    – Kay K.
    5 hours ago








2




2




$begingroup$
The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
$endgroup$
– Kay K.
5 hours ago





$begingroup$
The denominator in the 2nd line is $r^3sec^3theta$ instead of $r^3sec^6theta$.
$endgroup$
– Kay K.
5 hours ago











2 Answers
2






active

oldest

votes


















4












$begingroup$

You are doing $(rsectheta)^3=r^6sec^6theta$. Oops! ;-)




There's a slicker way to do it.



Get rid of the $r$ with $x=ru$ to begin with, so your integral becomes
$$
fracArint_a/r^inftyfrac1(1+u^2)^3/2,du
$$

Now let's concentrate on the antiderivative
$$
intfrac1(1+u^2)^3/2,du=
intfrac1+u^2-u^2(1+u^2)^3/2,du=
intfrac1(1+u^2)^1/2,du-intfracu^2(1+u^2)^3/2,du
$$

Do the second term by parts
$$
int ufracu(1+u^2)^3/2,du=
-fracu(1+u^2)^1/2+intfrac1(1+u^2)^1/2,du
$$

See what happens?
$$
intfrac1(1+u^2)^3/2,du=fracu(1+u^2)^1/2+c
$$

which we can verify by direct differentiation.



Now
$$
left[fracu(1+u^2)^1/2right]_a/r^infty=1-fraca/r(1+(a/r)^2)^1/2
=1-fraca(r^2+a^2)^1/2
$$

and your integral is indeed
$$
fracArleft(1-fracasqrtr^2+a^2right)
$$






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    Firstly you made an error in the first line of working
    $$(rsec(theta))^3=r^3sec^3(theta)$$
    Secondly, you need to change the range of integration after performing a substitution. If $theta=arctan(fracxr)$ then the limits should change as $x=a implies theta=arctan(fracar)$ also $x=infty implies theta=fracpi2$.






    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%2f3142908%2ftrig-subsitution-when-theres-no-square-root%23new-answer', 'question_page');

      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      You are doing $(rsectheta)^3=r^6sec^6theta$. Oops! ;-)




      There's a slicker way to do it.



      Get rid of the $r$ with $x=ru$ to begin with, so your integral becomes
      $$
      fracArint_a/r^inftyfrac1(1+u^2)^3/2,du
      $$

      Now let's concentrate on the antiderivative
      $$
      intfrac1(1+u^2)^3/2,du=
      intfrac1+u^2-u^2(1+u^2)^3/2,du=
      intfrac1(1+u^2)^1/2,du-intfracu^2(1+u^2)^3/2,du
      $$

      Do the second term by parts
      $$
      int ufracu(1+u^2)^3/2,du=
      -fracu(1+u^2)^1/2+intfrac1(1+u^2)^1/2,du
      $$

      See what happens?
      $$
      intfrac1(1+u^2)^3/2,du=fracu(1+u^2)^1/2+c
      $$

      which we can verify by direct differentiation.



      Now
      $$
      left[fracu(1+u^2)^1/2right]_a/r^infty=1-fraca/r(1+(a/r)^2)^1/2
      =1-fraca(r^2+a^2)^1/2
      $$

      and your integral is indeed
      $$
      fracArleft(1-fracasqrtr^2+a^2right)
      $$






      share|cite|improve this answer









      $endgroup$

















        4












        $begingroup$

        You are doing $(rsectheta)^3=r^6sec^6theta$. Oops! ;-)




        There's a slicker way to do it.



        Get rid of the $r$ with $x=ru$ to begin with, so your integral becomes
        $$
        fracArint_a/r^inftyfrac1(1+u^2)^3/2,du
        $$

        Now let's concentrate on the antiderivative
        $$
        intfrac1(1+u^2)^3/2,du=
        intfrac1+u^2-u^2(1+u^2)^3/2,du=
        intfrac1(1+u^2)^1/2,du-intfracu^2(1+u^2)^3/2,du
        $$

        Do the second term by parts
        $$
        int ufracu(1+u^2)^3/2,du=
        -fracu(1+u^2)^1/2+intfrac1(1+u^2)^1/2,du
        $$

        See what happens?
        $$
        intfrac1(1+u^2)^3/2,du=fracu(1+u^2)^1/2+c
        $$

        which we can verify by direct differentiation.



        Now
        $$
        left[fracu(1+u^2)^1/2right]_a/r^infty=1-fraca/r(1+(a/r)^2)^1/2
        =1-fraca(r^2+a^2)^1/2
        $$

        and your integral is indeed
        $$
        fracArleft(1-fracasqrtr^2+a^2right)
        $$






        share|cite|improve this answer









        $endgroup$















          4












          4








          4





          $begingroup$

          You are doing $(rsectheta)^3=r^6sec^6theta$. Oops! ;-)




          There's a slicker way to do it.



          Get rid of the $r$ with $x=ru$ to begin with, so your integral becomes
          $$
          fracArint_a/r^inftyfrac1(1+u^2)^3/2,du
          $$

          Now let's concentrate on the antiderivative
          $$
          intfrac1(1+u^2)^3/2,du=
          intfrac1+u^2-u^2(1+u^2)^3/2,du=
          intfrac1(1+u^2)^1/2,du-intfracu^2(1+u^2)^3/2,du
          $$

          Do the second term by parts
          $$
          int ufracu(1+u^2)^3/2,du=
          -fracu(1+u^2)^1/2+intfrac1(1+u^2)^1/2,du
          $$

          See what happens?
          $$
          intfrac1(1+u^2)^3/2,du=fracu(1+u^2)^1/2+c
          $$

          which we can verify by direct differentiation.



          Now
          $$
          left[fracu(1+u^2)^1/2right]_a/r^infty=1-fraca/r(1+(a/r)^2)^1/2
          =1-fraca(r^2+a^2)^1/2
          $$

          and your integral is indeed
          $$
          fracArleft(1-fracasqrtr^2+a^2right)
          $$






          share|cite|improve this answer









          $endgroup$



          You are doing $(rsectheta)^3=r^6sec^6theta$. Oops! ;-)




          There's a slicker way to do it.



          Get rid of the $r$ with $x=ru$ to begin with, so your integral becomes
          $$
          fracArint_a/r^inftyfrac1(1+u^2)^3/2,du
          $$

          Now let's concentrate on the antiderivative
          $$
          intfrac1(1+u^2)^3/2,du=
          intfrac1+u^2-u^2(1+u^2)^3/2,du=
          intfrac1(1+u^2)^1/2,du-intfracu^2(1+u^2)^3/2,du
          $$

          Do the second term by parts
          $$
          int ufracu(1+u^2)^3/2,du=
          -fracu(1+u^2)^1/2+intfrac1(1+u^2)^1/2,du
          $$

          See what happens?
          $$
          intfrac1(1+u^2)^3/2,du=fracu(1+u^2)^1/2+c
          $$

          which we can verify by direct differentiation.



          Now
          $$
          left[fracu(1+u^2)^1/2right]_a/r^infty=1-fraca/r(1+(a/r)^2)^1/2
          =1-fraca(r^2+a^2)^1/2
          $$

          and your integral is indeed
          $$
          fracArleft(1-fracasqrtr^2+a^2right)
          $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 5 hours ago









          egregegreg

          184k1486205




          184k1486205





















              3












              $begingroup$

              Firstly you made an error in the first line of working
              $$(rsec(theta))^3=r^3sec^3(theta)$$
              Secondly, you need to change the range of integration after performing a substitution. If $theta=arctan(fracxr)$ then the limits should change as $x=a implies theta=arctan(fracar)$ also $x=infty implies theta=fracpi2$.






              share|cite|improve this answer









              $endgroup$

















                3












                $begingroup$

                Firstly you made an error in the first line of working
                $$(rsec(theta))^3=r^3sec^3(theta)$$
                Secondly, you need to change the range of integration after performing a substitution. If $theta=arctan(fracxr)$ then the limits should change as $x=a implies theta=arctan(fracar)$ also $x=infty implies theta=fracpi2$.






                share|cite|improve this answer









                $endgroup$















                  3












                  3








                  3





                  $begingroup$

                  Firstly you made an error in the first line of working
                  $$(rsec(theta))^3=r^3sec^3(theta)$$
                  Secondly, you need to change the range of integration after performing a substitution. If $theta=arctan(fracxr)$ then the limits should change as $x=a implies theta=arctan(fracar)$ also $x=infty implies theta=fracpi2$.






                  share|cite|improve this answer









                  $endgroup$



                  Firstly you made an error in the first line of working
                  $$(rsec(theta))^3=r^3sec^3(theta)$$
                  Secondly, you need to change the range of integration after performing a substitution. If $theta=arctan(fracxr)$ then the limits should change as $x=a implies theta=arctan(fracar)$ also $x=infty implies theta=fracpi2$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 5 hours ago









                  Peter ForemanPeter Foreman

                  3,4521216




                  3,4521216



























                      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%2f3142908%2ftrig-subsitution-when-theres-no-square-root%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







                      -calculus, improper-integrals, integration, trigonometric-integrals

                      Popular posts from this blog

                      Frič See also Navigation menuinternal link

                      Identify plant with long narrow paired leaves and reddish stems Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?What is this plant with long sharp leaves? Is it a weed?What is this 3ft high, stalky plant, with mid sized narrow leaves?What is this young shrub with opposite ovate, crenate leaves and reddish stems?What is this plant with large broad serrated leaves?Identify this upright branching weed with long leaves and reddish stemsPlease help me identify this bulbous plant with long, broad leaves and white flowersWhat is this small annual with narrow gray/green leaves and rust colored daisy-type flowers?What is this chilli plant?Does anyone know what type of chilli plant this is?Help identify this plant

                      fontconfig warning: “/etc/fonts/fonts.conf”, line 100: unknown “element blank” The 2019 Stack Overflow Developer Survey Results Are In“tar: unrecognized option --warning” during 'apt-get install'How to fix Fontconfig errorHow do I figure out which font file is chosen for a system generic font alias?Why are some apt-get-installed fonts being ignored by fc-list, xfontsel, etc?Reload settings in /etc/fonts/conf.dTaking 30 seconds longer to boot after upgrade from jessie to stretchHow to match multiple font names with a single <match> element?Adding a custom font to fontconfigRemoving fonts from fontconfig <match> resultsBroken fonts after upgrading Firefox ESR to latest Firefox