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Math Help - integrate ln(tanx+1)

  1. #1
    Eater of Worlds
    galactus's Avatar
    Jul 2006
    Chaneysville, PA

    integrate ln(tanx+1)

    I seen this problem a while back on the site and to my recollection it wasn't tackled. If it was, my apologies. I couldn't find it. I think Isomorphism had posted it. If I am a for doing this, so be it.

    I am sure some of you have some clever show off methods, but here is something that kind of appeared to me rather quickly and turned into something familiar. I think it is cool to see the different innovative ways folks tackle these. It shows what a wonder the human brain can be.
    You can easily run them through a computer, but they lack the human touch.


    Let u=1+tan(x)



    Make the subs and get:


    An x mixed in with a u?. some that may seem like sacrilege.

    But it's OK:

    Since tan(x)=\frac{u-1}{1}

    Then sec^{2}(u)=(u-1)^{2}+1

    Then we have:


    Now, let w=u-1, \;\ w+1=u, \;\ dw=du

    Then we have:


    Now...this is a familiar integral that the Krizmeister done awhile back so I won't bother.

    It's solution is \frac{\pi}{8}ln(2)

    This kind of popped into my head when I seen it so I had to write it down.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Mar 2007
    Santiago, Chile
    Here. (That link will direct you into another one.)
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