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Math Help - integrate ln(tanx+1)

  1. #1
    Eater of Worlds
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    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.

    \int_{0}^{\frac{\pi}{4}}ln(tan(x)+1)dx

    Let u=1+tan(x)

    u-1=tan(x)dx

    du=sec^{2}(x)dx

    Make the subs and get:

    \int\frac{ln(u)}{sec^{2}(x)}dx

    An x mixed in with a u?. No.......to 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:

    \int_{1}^{2}\frac{ln(u)}{(u-1)^{2}+1}du

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

    Then we have:

    \int_{0}^{1}\frac{ln(w+1)}{w^{2}+1}dw

    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
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