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Improper Integral of theta/cos^2 theta

  1. Dec 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Improper Integral of theta/cos^2 theta

    2. Relevant equations


    3. The attempt at a solution

    Hi all, this was one of the few questions on my final today that I just didn't know how to do. I know how to do trig sub, know all my trig identities and know improper integration, but was a bit at a loss for this one.

    I could use a half angle for the denominator --> theta / 1/2 [1 + cos(2 theta)] -->

    Maybe integrate, and get theta^2 / 1/2 theta + 1/2(sin 2 theta).

    I'm sure what I tried was very wrong, but I wanted to make some kind of attempt.

    Edit: nevermind, you can't integrate like that.
     
    Last edited: Dec 17, 2014
  2. jcsd
  3. Dec 17, 2014 #2

    Mark44

    Staff: Mentor

    $$\int \frac{\theta d\theta}{cos^2(\theta)} = \int \theta sec^2(\theta) d\theta$$

    Use integration by parts with a judicious choice for u and dv.
     
  4. Dec 17, 2014 #3
    Damn, that's a pretty easy integration by parts question actually. So, if I get an answer of tan(theta) - ln(sec(theta)), where would the improper integration come into play?

    Oh wait, natural log functions must be greater than zero. So, it would be something like, the limit, as b approaches 0, from the right, of tan(theta) - ln(sec(theta))?
     
  5. Dec 17, 2014 #4

    Mark44

    Staff: Mentor

    It's not a hard integration by parts, but the answer you show is incorrect. If you differentiate your answer, you don't get ##\theta sec^2(\theta)##.
    To be more precise, the argument of a log function must be greater than zero. The output of a log function can be any real number.

    The integral you showed was an indefinite integral. An improper integral is a definite integral for which the integrand is undefined at one or more points inside the interval defined by the limits of integration, or at one or both endpoints.
     
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