Question about integral of an odd functionHow can I find the integral of this function using trig...

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Question about integral of an odd function


How can I find the integral of this function using trig substitution?Why does $sin{alpha}cdot isin{alpha x}$ disappear from this integral?Using Polar Coordinates to Calculate Double IntegralQuestion concerning the domain of polar coordinate.Evaluate $int_0^{infty}frac{e^{-x}-e^{-2x}}{x}dx$ using a double integralQuestion about the limits of definite integralsQuestion about a substitution in an integralDoubt about an improper multiple integralIntegrate $int_0^1 sin^{-1}{frac{x^2}{1+x^2}}dx$Studying the convergence of the integral $int_0^pi frac{ln(sin(x))}{x}dx$













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I am studying something and encountered this:

"
Let $R(theta,T) = int_{-T}^{T} frac{(sin theta t)}{t}dt, S(T) = int_0^Tfrac{(sin x)}{x}dx$, then for $theta > 0$ and changing variables $t=x/theta $ shows that



$R(theta,T)=2int_0^{Ttheta}frac{sin x}{x}dx = 2S(Ttheta)$ while for $theta<0$, $R(theta,T) = -R(|theta|,T)$" which I don't understand.



If $theta<0$, then $R(theta,T)=2int_{-T|theta|}^{0}frac{sin x}{x}dx =2int_0^{T|theta|}frac{sin x}{x}dx = R(|theta|,T)$ as $frac{sin x}{x}$ is an even function, right? I am missing something simple here, thanks and appreciate an explanation.










share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    I am studying something and encountered this:

    "
    Let $R(theta,T) = int_{-T}^{T} frac{(sin theta t)}{t}dt, S(T) = int_0^Tfrac{(sin x)}{x}dx$, then for $theta > 0$ and changing variables $t=x/theta $ shows that



    $R(theta,T)=2int_0^{Ttheta}frac{sin x}{x}dx = 2S(Ttheta)$ while for $theta<0$, $R(theta,T) = -R(|theta|,T)$" which I don't understand.



    If $theta<0$, then $R(theta,T)=2int_{-T|theta|}^{0}frac{sin x}{x}dx =2int_0^{T|theta|}frac{sin x}{x}dx = R(|theta|,T)$ as $frac{sin x}{x}$ is an even function, right? I am missing something simple here, thanks and appreciate an explanation.










    share|cite|improve this question









    $endgroup$















      3












      3








      3





      $begingroup$


      I am studying something and encountered this:

      "
      Let $R(theta,T) = int_{-T}^{T} frac{(sin theta t)}{t}dt, S(T) = int_0^Tfrac{(sin x)}{x}dx$, then for $theta > 0$ and changing variables $t=x/theta $ shows that



      $R(theta,T)=2int_0^{Ttheta}frac{sin x}{x}dx = 2S(Ttheta)$ while for $theta<0$, $R(theta,T) = -R(|theta|,T)$" which I don't understand.



      If $theta<0$, then $R(theta,T)=2int_{-T|theta|}^{0}frac{sin x}{x}dx =2int_0^{T|theta|}frac{sin x}{x}dx = R(|theta|,T)$ as $frac{sin x}{x}$ is an even function, right? I am missing something simple here, thanks and appreciate an explanation.










      share|cite|improve this question









      $endgroup$




      I am studying something and encountered this:

      "
      Let $R(theta,T) = int_{-T}^{T} frac{(sin theta t)}{t}dt, S(T) = int_0^Tfrac{(sin x)}{x}dx$, then for $theta > 0$ and changing variables $t=x/theta $ shows that



      $R(theta,T)=2int_0^{Ttheta}frac{sin x}{x}dx = 2S(Ttheta)$ while for $theta<0$, $R(theta,T) = -R(|theta|,T)$" which I don't understand.



      If $theta<0$, then $R(theta,T)=2int_{-T|theta|}^{0}frac{sin x}{x}dx =2int_0^{T|theta|}frac{sin x}{x}dx = R(|theta|,T)$ as $frac{sin x}{x}$ is an even function, right? I am missing something simple here, thanks and appreciate an explanation.







      calculus






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      asked 4 hours ago









      manifoldedmanifolded

      3487




      3487






















          2 Answers
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          4












          $begingroup$

          When $t=-T$ we get $x=ttheta =-Ttheta =T|theta|$ and not $-T|theta|$.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Correct! Thanks.
            $endgroup$
            – manifolded
            4 hours ago



















          2












          $begingroup$

          $$begin{align}theta <0 & Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (theta t)}{t}dt\
          &Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (-|theta| t)}{t}dt\
          &Rightarrow R(theta ,T)=-int_{-T}^Tfrac{sin (|theta| t)}{t}dt [because sin(-x)=-sin (x)big]\
          &Rightarrow R(theta ,T)=-R(|theta|,T)\
          end{align}$$






          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            2 Answers
            2






            active

            oldest

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            active

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            active

            oldest

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            4












            $begingroup$

            When $t=-T$ we get $x=ttheta =-Ttheta =T|theta|$ and not $-T|theta|$.






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              Correct! Thanks.
              $endgroup$
              – manifolded
              4 hours ago
















            4












            $begingroup$

            When $t=-T$ we get $x=ttheta =-Ttheta =T|theta|$ and not $-T|theta|$.






            share|cite|improve this answer









            $endgroup$









            • 1




              $begingroup$
              Correct! Thanks.
              $endgroup$
              – manifolded
              4 hours ago














            4












            4








            4





            $begingroup$

            When $t=-T$ we get $x=ttheta =-Ttheta =T|theta|$ and not $-T|theta|$.






            share|cite|improve this answer









            $endgroup$



            When $t=-T$ we get $x=ttheta =-Ttheta =T|theta|$ and not $-T|theta|$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 4 hours ago









            Kavi Rama MurthyKavi Rama Murthy

            63.1k42362




            63.1k42362








            • 1




              $begingroup$
              Correct! Thanks.
              $endgroup$
              – manifolded
              4 hours ago














            • 1




              $begingroup$
              Correct! Thanks.
              $endgroup$
              – manifolded
              4 hours ago








            1




            1




            $begingroup$
            Correct! Thanks.
            $endgroup$
            – manifolded
            4 hours ago




            $begingroup$
            Correct! Thanks.
            $endgroup$
            – manifolded
            4 hours ago











            2












            $begingroup$

            $$begin{align}theta <0 & Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (theta t)}{t}dt\
            &Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (-|theta| t)}{t}dt\
            &Rightarrow R(theta ,T)=-int_{-T}^Tfrac{sin (|theta| t)}{t}dt [because sin(-x)=-sin (x)big]\
            &Rightarrow R(theta ,T)=-R(|theta|,T)\
            end{align}$$






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              $$begin{align}theta <0 & Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (theta t)}{t}dt\
              &Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (-|theta| t)}{t}dt\
              &Rightarrow R(theta ,T)=-int_{-T}^Tfrac{sin (|theta| t)}{t}dt [because sin(-x)=-sin (x)big]\
              &Rightarrow R(theta ,T)=-R(|theta|,T)\
              end{align}$$






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                $$begin{align}theta <0 & Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (theta t)}{t}dt\
                &Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (-|theta| t)}{t}dt\
                &Rightarrow R(theta ,T)=-int_{-T}^Tfrac{sin (|theta| t)}{t}dt [because sin(-x)=-sin (x)big]\
                &Rightarrow R(theta ,T)=-R(|theta|,T)\
                end{align}$$






                share|cite|improve this answer









                $endgroup$



                $$begin{align}theta <0 & Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (theta t)}{t}dt\
                &Rightarrow R(theta ,T)=int_{-T}^Tfrac{sin (-|theta| t)}{t}dt\
                &Rightarrow R(theta ,T)=-int_{-T}^Tfrac{sin (|theta| t)}{t}dt [because sin(-x)=-sin (x)big]\
                &Rightarrow R(theta ,T)=-R(|theta|,T)\
                end{align}$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 4 hours ago









                s0ulr3aper07s0ulr3aper07

                541111




                541111






























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