Results 1 to 2 of 2

Thread: 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.

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

    Let $\displaystyle u=1+tan(x)$

    $\displaystyle u-1=tan(x)dx$

    $\displaystyle du=sec^{2}(x)dx$

    Make the subs and get:

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

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

    But it's OK:

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

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

    Then we have:

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

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

    Then we have:

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

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

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

    This kind of popped into my head when I seen it so I had to write it down.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Math Engineering Student
    Krizalid's Avatar
    Mar 2007
    Santiago, Chile
    Here. (That link will direct you into another one.)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. How to integrate squareroot of tanx
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Aug 27th 2011, 09:05 AM
  2. Prove:(tanx/(1+tanx)) = (sinx/(sinx+cosx))
    Posted in the Trigonometry Forum
    Replies: 7
    Last Post: Feb 2nd 2011, 08:46 PM
  3. integrate of sqrt tanx
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jul 2nd 2010, 10:47 PM
  4. tanx=1.35
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: Feb 4th 2010, 09:22 PM
  5. sinx + tanx > 2x
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: Mar 6th 2008, 02:14 PM

Search tags for this page

Search Tags

/mathhelpforum @mathhelpforum