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Integral: 1/(1+cos(x)+sin(x)) help

  1. Jan 14, 2010 #1
    Can someone help me with this integral?

    Integral: 1/(1+cos(x)+sin(x))
     
    Last edited: Jan 14, 2010
  2. jcsd
  3. Jan 14, 2010 #2

    D H

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    Re: Integrals

    Solving this will take some adroit trig substitutions. Half angle formulae, particularly that for tan(x/2), will come in handy here.
     
  4. Jan 14, 2010 #3
    Re: Integrals

    I've tried it, but my answer isn't correct..

    cos(x) = (1 - (tan(x/2))^2) / (1 + (tan(x/2))^2)

    sin(x) = (2*tan(x/2)) / (1 + (tan(x/2))^2)

    And t = tan(x/2) ==> x = 2 bgtan(t) ==> dx = 2/(1+x^2)

    If I do it like that i become this integral:

    1/(t+1) dt

    But that's not correct. Have I done something wrong?
     
  5. Jan 14, 2010 #4

    D H

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    Re: Integrals

    That is exactly right, but you haven't finished yet. Continue on!
     
  6. Jan 14, 2010 #5
    Re: Integrals

    Than I get

    ln (t+1)

    = ln (tan(x/2) +1)

    = ln ((sin(x/2) / cos(x/2)) +1)

    = ln (sin(x/2) +1) - ln(cos(x/2)+1)

    And what now?
     
  7. Jan 14, 2010 #6
    Re: Integrals

    Than I get

    ln (t+1)

    = ln (tan(x/2) +1)

    = ln ((sin(x/2) / cos(x/2)) +1)

    = ln (sin(x/2) +1) - ln(cos(x/2)+1)

    And what now?
     
  8. Jan 14, 2010 #7

    D H

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    Re: Integrals

    Good so far.

    Whoa! What's this last step?

    For that matter, why do you need to go beyond ln(tan(x/2)+1) ? That is a perfectly good answer in and of itself.
     
  9. Jan 14, 2010 #8
    Re: Integrals

    Ow, I'm verry sorry.. I made a mistake, I read the wrong answer in my book.. ln (tan(x/2) +1) is the correct answer..
     
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