How can prove this integralHow to calculate the derivative of this integral?how prove this integral...

Does dispel magic end a master's control over their undead?

How do you voice extended chords?

How to use Mathematica to do a complex integrate with poles in real axis?

General past possibility with 'could'

What is the wife of a henpecked husband called?

Do authors have to be politically correct in article-writing?

ST_Buffer in PostGIS produces different results for the same set of lines

How can prove this integral

Which communication protocol is used in AdLib sound card?

What is the difference between rolling more dice versus fewer dice?

Why am I able to open Wireshark in macOS X without root privileges?

Play Zip, Zap, Zop

Picture with grey box as background

Citing paywalled articles accessed via illegal web sharing

How to deal with possible delayed baggage?

Why is Agricola named as such?

Crontab: Ubuntu running script (noob)

Clues on how to solve these types of problems within 2-3 minutes for competitive exams

"on its way" vs. "in its way"

When can a QA tester start his job?

A starship is travelling at 0.9c and collides with a small rock. Will it leave a clean hole through, or will more happen?

Has any human ever had the choice to leave Earth permanently?

Is it a fallacy if someone claims they need an explanation for every word of your argument to the point where they don't understand common terms?

Why was Lupin comfortable with saying Voldemort's name?



How can prove this integral


How to calculate the derivative of this integral?how prove this integral inequality?Could anybody check this integral?Help! How to solve this integral?Can anyone help me with this improper integral?Any idea how to solve this integral?How can I integrate this? Improper Integral.Integral smaller than $frac{1}{2}epsilon$Stochastic Geometry : Obtaining an IntegralHow would one prove the existence of the following indefinite integral













4












$begingroup$


I was reading a textbook which these two equations posed . The second one was the result of the first one .
How can we say that ?



If we know :
$$int_{0^+}^{+infty} frac{sin(x)}{x} = frac{pi}{2}$$



How can we prove :
$$int_{0^+}^{+infty} left(frac{sin(x)}{x}right)^2 = frac{pi}{2}$$



Thanks in advance






real-analysis calculus integration







share|cite|improve this question









New contributor




Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$

















    4












    $begingroup$


    I was reading a textbook which these two equations posed . The second one was the result of the first one .
    How can we say that ?



    If we know :
    $$int_{0^+}^{+infty} frac{sin(x)}{x} = frac{pi}{2}$$



    How can we prove :
    $$int_{0^+}^{+infty} left(frac{sin(x)}{x}right)^2 = frac{pi}{2}$$



    Thanks in advance






    real-analysis calculus integration







    share|cite|improve this question









    New contributor




    Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$















      4












      4








      4


      1



      $begingroup$


      I was reading a textbook which these two equations posed . The second one was the result of the first one .
      How can we say that ?



      If we know :
      $$int_{0^+}^{+infty} frac{sin(x)}{x} = frac{pi}{2}$$



      How can we prove :
      $$int_{0^+}^{+infty} left(frac{sin(x)}{x}right)^2 = frac{pi}{2}$$



      Thanks in advance






      real-analysis calculus integration







      share|cite|improve this question









      New contributor




      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      I was reading a textbook which these two equations posed . The second one was the result of the first one .
      How can we say that ?



      If we know :
      $$int_{0^+}^{+infty} frac{sin(x)}{x} = frac{pi}{2}$$



      How can we prove :
      $$int_{0^+}^{+infty} left(frac{sin(x)}{x}right)^2 = frac{pi}{2}$$



      Thanks in advance






      real-analysis calculus integration




      real-analysis calculus integration






      share|cite|improve this question









      New contributor




      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited 5 hours ago









      Alan Muniz

      2,2711829




      2,2711829






      New contributor




      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked 6 hours ago









      RezaReza

      233




      233




      New contributor




      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Reza is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






















          2 Answers
          2






          active

          oldest

          votes


















          6












          $begingroup$

          Use $int_{0}^{infty} frac{sin(x)}{x} = frac{pi}{2}$ and $sin (2x)= 2 sin(x) cos(x)$ to get



          $$ (*) quadint_{0}^{infty} frac{sin(x) cos (x)}{x} =frac{pi}{4}.$$



          Then use integration by parts in $(*)$ to derive



          $$int_{0}^{infty} frac{sin^2(x)}{x^2} = frac{pi}{2}.$$






          share|cite|improve this answer









          $endgroup$









          • 2




            $begingroup$
            I can't figure it out how you derive from (*) to answer
            $endgroup$
            – Reza
            5 hours ago










          • $begingroup$
            Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
            $endgroup$
            – FDP
            3 hours ago





















          2












          $begingroup$

          $$I(a)=int_{-infty}^{+infty}dfrac{sin^2ax}{x^2}mathrm dx\ dfrac{mathrm dI}{mathrm da}=int_{-infty}^{+infty}partial_a dfrac{sin^2ax}{x^2}mathrm dx=2int_{0}^{infty}dfrac{sin 2ax}{x}mathrm dx=pi\ I(a)=pi a implies int_{0}^{infty}dfrac{sin^2x}{x^2}mathrm dx =dfrac{1}{2}I(1)=dfrac{pi}{2}$$






          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
            });


            }
            });






            Reza is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3128751%2fhow-can-prove-this-integral%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









            6












            $begingroup$

            Use $int_{0}^{infty} frac{sin(x)}{x} = frac{pi}{2}$ and $sin (2x)= 2 sin(x) cos(x)$ to get



            $$ (*) quadint_{0}^{infty} frac{sin(x) cos (x)}{x} =frac{pi}{4}.$$



            Then use integration by parts in $(*)$ to derive



            $$int_{0}^{infty} frac{sin^2(x)}{x^2} = frac{pi}{2}.$$






            share|cite|improve this answer









            $endgroup$









            • 2




              $begingroup$
              I can't figure it out how you derive from (*) to answer
              $endgroup$
              – Reza
              5 hours ago










            • $begingroup$
              Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
              $endgroup$
              – FDP
              3 hours ago


















            6












            $begingroup$

            Use $int_{0}^{infty} frac{sin(x)}{x} = frac{pi}{2}$ and $sin (2x)= 2 sin(x) cos(x)$ to get



            $$ (*) quadint_{0}^{infty} frac{sin(x) cos (x)}{x} =frac{pi}{4}.$$



            Then use integration by parts in $(*)$ to derive



            $$int_{0}^{infty} frac{sin^2(x)}{x^2} = frac{pi}{2}.$$






            share|cite|improve this answer









            $endgroup$









            • 2




              $begingroup$
              I can't figure it out how you derive from (*) to answer
              $endgroup$
              – Reza
              5 hours ago










            • $begingroup$
              Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
              $endgroup$
              – FDP
              3 hours ago
















            6












            6








            6





            $begingroup$

            Use $int_{0}^{infty} frac{sin(x)}{x} = frac{pi}{2}$ and $sin (2x)= 2 sin(x) cos(x)$ to get



            $$ (*) quadint_{0}^{infty} frac{sin(x) cos (x)}{x} =frac{pi}{4}.$$



            Then use integration by parts in $(*)$ to derive



            $$int_{0}^{infty} frac{sin^2(x)}{x^2} = frac{pi}{2}.$$






            share|cite|improve this answer









            $endgroup$



            Use $int_{0}^{infty} frac{sin(x)}{x} = frac{pi}{2}$ and $sin (2x)= 2 sin(x) cos(x)$ to get



            $$ (*) quadint_{0}^{infty} frac{sin(x) cos (x)}{x} =frac{pi}{4}.$$



            Then use integration by parts in $(*)$ to derive



            $$int_{0}^{infty} frac{sin^2(x)}{x^2} = frac{pi}{2}.$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 5 hours ago









            FredFred

            47k1848




            47k1848








            • 2




              $begingroup$
              I can't figure it out how you derive from (*) to answer
              $endgroup$
              – Reza
              5 hours ago










            • $begingroup$
              Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
              $endgroup$
              – FDP
              3 hours ago
















            • 2




              $begingroup$
              I can't figure it out how you derive from (*) to answer
              $endgroup$
              – Reza
              5 hours ago










            • $begingroup$
              Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
              $endgroup$
              – FDP
              3 hours ago










            2




            2




            $begingroup$
            I can't figure it out how you derive from (*) to answer
            $endgroup$
            – Reza
            5 hours ago




            $begingroup$
            I can't figure it out how you derive from (*) to answer
            $endgroup$
            – Reza
            5 hours ago












            $begingroup$
            Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
            $endgroup$
            – FDP
            3 hours ago






            $begingroup$
            Reza: What is an antiderivative of the function cos? Can you derive the function defined for $x> 0$ by $f(x)=frac{sin x}{x}$ (product of functions)?
            $endgroup$
            – FDP
            3 hours ago













            2












            $begingroup$

            $$I(a)=int_{-infty}^{+infty}dfrac{sin^2ax}{x^2}mathrm dx\ dfrac{mathrm dI}{mathrm da}=int_{-infty}^{+infty}partial_a dfrac{sin^2ax}{x^2}mathrm dx=2int_{0}^{infty}dfrac{sin 2ax}{x}mathrm dx=pi\ I(a)=pi a implies int_{0}^{infty}dfrac{sin^2x}{x^2}mathrm dx =dfrac{1}{2}I(1)=dfrac{pi}{2}$$






            share|cite|improve this answer











            $endgroup$


















              2












              $begingroup$

              $$I(a)=int_{-infty}^{+infty}dfrac{sin^2ax}{x^2}mathrm dx\ dfrac{mathrm dI}{mathrm da}=int_{-infty}^{+infty}partial_a dfrac{sin^2ax}{x^2}mathrm dx=2int_{0}^{infty}dfrac{sin 2ax}{x}mathrm dx=pi\ I(a)=pi a implies int_{0}^{infty}dfrac{sin^2x}{x^2}mathrm dx =dfrac{1}{2}I(1)=dfrac{pi}{2}$$






              share|cite|improve this answer











              $endgroup$
















                2












                2








                2





                $begingroup$

                $$I(a)=int_{-infty}^{+infty}dfrac{sin^2ax}{x^2}mathrm dx\ dfrac{mathrm dI}{mathrm da}=int_{-infty}^{+infty}partial_a dfrac{sin^2ax}{x^2}mathrm dx=2int_{0}^{infty}dfrac{sin 2ax}{x}mathrm dx=pi\ I(a)=pi a implies int_{0}^{infty}dfrac{sin^2x}{x^2}mathrm dx =dfrac{1}{2}I(1)=dfrac{pi}{2}$$






                share|cite|improve this answer











                $endgroup$



                $$I(a)=int_{-infty}^{+infty}dfrac{sin^2ax}{x^2}mathrm dx\ dfrac{mathrm dI}{mathrm da}=int_{-infty}^{+infty}partial_a dfrac{sin^2ax}{x^2}mathrm dx=2int_{0}^{infty}dfrac{sin 2ax}{x}mathrm dx=pi\ I(a)=pi a implies int_{0}^{infty}dfrac{sin^2x}{x^2}mathrm dx =dfrac{1}{2}I(1)=dfrac{pi}{2}$$







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 5 hours ago

























                answered 5 hours ago









                Paras KhoslaParas Khosla

                1,384219




                1,384219






















                    Reza is a new contributor. Be nice, and check out our Code of Conduct.










                    draft saved

                    draft discarded


















                    Reza is a new contributor. Be nice, and check out our Code of Conduct.













                    Reza is a new contributor. Be nice, and check out our Code of Conduct.












                    Reza is a new contributor. Be nice, and check out our Code of Conduct.
















                    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%2f3128751%2fhow-can-prove-this-integral%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

                    Why not use the yoke to control yaw, as well as pitch and roll? Announcing the arrival of...

                    Image
                    Does the Mueller report show a conspiracy between Russia and the Trump Campaign? Relating to the President and obstruction, were Mueller's conclusions preordained? How much damage would a cupful of neutron star matter do to the Earth? Nose gear failure in single prop aircraft: belly landing or nose landing? Should a wizard buy fine inks every time he want to copy spells into his spellbook? Can you force honesty by using the Speak with Dead and Zone of Truth spells together? A trigonometry question from STEP examination Special flights What is the "studentd" process? systemd and copy (/bin/cp): no such file or directory...
                    Read more

                    Couldn't open a raw socket. Error: Permission denied (13) (nmap)Is it possible to run networking commands...

                    Image
                    Why didn't Boeing produce its own regional jet? What is the most common color to indicate the input-field is disabled? Theorists sure want true answers to this! What is the opposite of "eschatology"? What do you call someone who asks many questions? Is there a hemisphere-neutral way of specifying a season? Is this draw by repetition? Bug? String.split chopping off empty trailing value Is it inappropriate for a student to attend their mentor's dissertation defense? How dangerous is XSS Why do I get negative height? Is it possible to map ...
                    Read more

                    VNC viewer RFB protocol error: bad desktop size 0x0I Cannot Type the Key 'd' (lowercase) in VNC Viewer...

                    Image
                    risk of flooding in petra in november How to get directions in deep space? How could a planet have erratic days? 15% tax on $7.5k earnings. Is that right? Did the UK lift the requirement for registering SIM cards? Is it allowed to activate the ability of multiple planeswalkers in a single turn? Stack Interview Code methods made from class Node and Smart Pointers Non-trope happy ending? What is the highest possible scrabble score for placing a single tile Open a doc from terminal, but not by its name The IT department bottlenecks progress, how should I handle this? ...
                    Read more