1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted