Complexifying the integral of the secant function

[Wisc17]

Since learning about being able to complexify differential equations (I am doing the MIT OCW course by Arthur Mattuck), I have tried to apply to this to particular problems in integration as well. Specifically, I have tried to integrate the secant of some function to see if it would lead to the same accepted answer as when you used trignometric identities and u substitutions. What I have done is:


∫secΘdΘ = ∫(1/cosΘ)dΘ = Re{∫(1/e^iΘ)dΘ}


∫(1/e^iΘ)dΘ = ∫e^-iΘdΘ = -e^iΘ/i = -1/(i(cosΘ+isinΘ))

= -1/(-sin+icosΘ) = -1(-sin-icosΘ)/(sin²-(-cos²Θ)) = sinΘ+icosΘ

Re(sinΘ+icosΘ) = sinΘ

I know sinΘ is not the correct answer, but I do not understand why I cannot do the math this way. In the video I watched for complexifying integrals, I watched the professor do:

∫cosΘdΘ = Re{∫e^iΘdΘ}

and the professor got the correct answer doing this, so I do not understand why I cannot do this with the secant.

I have limited access to teachers right now, but I did visit one teacher at a local college. He said I cannot complexify the equation as I have done, rather that I should have done this:

cosΘ = (e^iΘ+e^-iΘ)/2

However, he did not explain why, and before I could ask he went off and explained how to do the integral in the "traditional way." I am not interested in the traditional way; I am trying to find another way to do the integral, and am trying to find the hole in my knowledge regarding complexifying problems.

Any help would be appreciated. Thank you.

 

[mfb]

∫(1/cosΘ)dΘ = Re{∫(1/e^iΘ)dΘ}

Why do you expect this to be true?

I see this step:
∫(1/cosΘ)dΘ = ∫(1/Re(e^iΘ))dΘ

What did you do afterwards?