How do we find the real part of the integral of sec(x) sec(x)?

[cragar]

<br /> sec(x) = \frac{2}{e^{ix}+e^{-ix}}<br />
then i multply bot top and bottom by e^{ix}
so i can do a u substitution
u=e^{ix} du=ie^{ix}
so then \int {\frac{2du}{(u^2+1)i}} <br /> =\frac {2arctan(u)}{i}}
so then i turn the arctan into a log
then i get ln|e^{ix}+i|-ln|e^{ix}-i| + c
then how do i get the real part out if this .

 

[arildno]

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

You'll get a rational integrand in u that you can solve by partial fractions decomposition:
\int\sec(x)dx=\int\frac{2du}{1-u^{2}}

 

[cragar]

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
\frac{cos(x)}{1-(sin(x))^2}
then u=sin(x) and du=cos(x)

 

[Gib Z]

Combine the two log terms into one, and use Log z = \ln |z| + i Arg(z). Ie the Real part is simply the natural log of the modulus.

 

[cragar]

thanks for all of your answers guys , I am 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 .

 

[Gib Z]

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.

 

[arildno]

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.

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

 

[Gib Z]

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. =]