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How to solve this definite integral?

  1. May 1, 2014 #1

    PhysicoRaj

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    1. The problem statement, all variables and given/known data

    Integrate:[tex]I=\int_{-π/4}^{π/4} \ln{(\sec θ-\tan θ)}\,dθ[/tex]

    2. Relevant equations

    Properties of definite integrals, basic integration formulae, trigonometric identities.

    3. The attempt at a solution
    By properties of definite integrals, the same integral I wrote as equivalent to[tex]I=\int_{-π/4}^{π/4} \ln{(\sec θ+\tan θ)}\,dθ[/tex].
    Because[tex]\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(a+b-x)\,dx[/tex](replacing θ by π/4-π/4-θ) Now, I think of adding these two integrals to form an equation and solving for [itex]I[/itex] but I'm messing up. Am I doing wrong? Is there any better/easy way?
    Thanks for your time and help.
     
  2. jcsd
  3. May 1, 2014 #2
    What do you get if you add them? It's quite obvious.

    Instead, you can notice that the integrand is an odd function, do you see why?
     
  4. May 1, 2014 #3

    PhysicoRaj

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    I tried that at first but I couldn't come to a decision, I can't see if it's odd or even. The secant is an even function right?
     
  5. May 1, 2014 #4
    Yes, secant is an even function.

    Define ##f(\theta)=\ln\left(\sec\theta-\tan\theta\right)##. Hence, ##f(-\theta)=\ln\left(\sec\theta+\tan\theta\right)##. What do you get if you add ##f(\theta)## and ##f(-\theta)##? Please show the attempt.
     
  6. May 1, 2014 #5

    PhysicoRaj

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    It's [itex]ln(1)=0[/itex]
     
  7. May 1, 2014 #6
    Right!

    So ##f(\theta)+f(-\theta)=0##. Do you see why it is an odd function? :)
     
  8. May 1, 2014 #7

    PhysicoRaj

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    Ahh.. I get it right now Pranav, thanks a lot! :smile:
     
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