Integral of sec(x)

  1. [tex]
    sec(x) = \frac{2}{e^{ix}+e^{-ix}}
    [/tex]
    then i multply bot top and bottom by [tex] e^{ix} [/tex]
    so i can do a u substitution
    [tex] u=e^{ix} du=ie^{ix} [/tex]
    so then [tex] \int {\frac{2du}{(u^2+1)i}}
    =\frac {2arctan(u)}{i}} [/tex]
    so then i turn the arctan into a log
    then i get [tex] ln|e^{ix}+i|-ln|e^{ix}-i| + c [/tex]
    then how do i get the real part out if this .
     
    Last edited: Jun 13, 2010
  2. jcsd
  3. arildno

    arildno 12,015
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    Gold Member

    Well, you MIGHT do it that way, but a simpler integration would be to set:
    [tex]1=\cos^2(\frac{x}{2})+\sin^{2}\frac{x}{2}[/tex]
    [tex]\cos(x)=\cos^{2}(\frac{x}{2})-\sin^{2}(\frac{x}{2})[/tex]
    These identities implies:
    [tex]\sec(x)=\frac{1+\tan^{2}(\frac{x}{2})}{1-\tan^{2}(\frac{x}{2})}[/tex]
    Setting, therefore:
    [tex]u=\tan(\frac{x}{2})\to\frac{du}{dx}=\frac{1}{2}\sec^{2}(\frac{x}{2})=\frac{1}{2}(1+u^{2})[/tex]

    You'll get a rational integrand in u that you can solve by partial fractions decomposition:
    [tex]\int\sec(x)dx=\int\frac{2du}{1-u^{2}}[/tex]
     
  4. sorry i should have said i want to see it done with complex numbers ,
    I have done it that way before . but i wrote it like
    [tex] \frac{cos(x)}{1-(sin(x))^2} [/tex]
    then u=sin(x) and du=cos(x)
     
  5. Gib Z

    Gib Z 3,348
    Homework Helper

    Combine the two log terms into one, and use [tex]Log z = \ln |z| + i Arg(z)[/tex]. Ie the Real part is simply the natural log of the modulus.
     
  6. thanks for all of your answers guys , im not sure what modulus is i tired looking it up
    could you maybe tell me where to read about it i have only had calc 3 .
     
  7. Gib Z

    Gib Z 3,348
    Homework Helper

    I'm sure you have if your doing integration like this! The modulus of a complex number a+bi is sqrt(a^2+b^2). You can think of it as the length of the line that connects the origin to a+bi on the Argand Plane.
     
  8. arildno

    arildno 12,015
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    Gold Member

    Wessel Plane, if I may.
    http://en.wikipedia.org/wiki/Caspar_Wessel
     
  9. Gib Z

    Gib Z 3,348
    Homework Helper

    Ahh my mistake !

    In mathematics often things aren't named after who really should have gotten credit for them! There's a joke that for an entire century after Euler, to ensure other mathematicians got some recognition, things were named after the first person after Euler to discover it. =]
     
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