1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral of ln(secx + tanx)?

  1. Mar 12, 2008 #1
    I was taking the integral of the secant function. Twice...
    The first one is simple, but what is the integral of
    ln(secx + tanx)dx?

    I've tried a few things, the first being integration by parts with u = ln(secx + tanx+) and dv = dx
    This just cancels in the end to 0 = 0
    I also rewrote it as int[ln(1+sinx) - ln(cosx)]dx but that doesn't seem to be any easier.

    Any suggestion would be greatly appreciated.
    Last edited: Mar 12, 2008
  2. jcsd
  3. Mar 12, 2008 #2


    User Avatar

    It doesn't have a closed form answer in terms of elementary functions.
  4. Mar 12, 2008 #3
    sure it does, unless you consider the polylog erudite:

    \frac{1}{2} \left(\pi \left(i x+\text{Log}\left[-\text{Sin}\left[\frac{1}{4} (\pi -2 x)\right]\right]-\text{Log}[1+i \text{Cos}[x]-\text{Sin}[x]]-\text{Log}[1-i
    \text{Cos}[x]+\text{Sin}[x]]+\text{Log}\left[\text{Sin}\left[\frac{1}{4} (\pi +2 x)\right]\right]\right)+2 x \left(2 i \text{ArcTan}\left[e^{i x}\right]+\text{Log}[\text{Sec}[x]+\text{Tan}[x]]\right)+2
    i \text{PolyLog}\left[2,i e^{i x}\right]-2 i \text{PolyLog}[2,-i \text{Cos}[x]+\text{Sin}[x]]\right)[/tex]

    lol off screen. whatever it's in the body of the post, just click quote or something
    Last edited: Mar 12, 2008
  5. Mar 12, 2008 #4


    User Avatar

    Elementary functions are finite combinations and compositions of algebraic, logarithmic, and exponential functions. Polylog obviously doesn't fit that description.
  6. Mar 12, 2008 #5
    yea you're right
  7. Jan 12, 2011 #6
    @ice109: How do you get that formula? (The one that involves PolyLog)
  8. Jan 12, 2011 #7
  9. Jan 13, 2011 #8
    Thank you!!!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Integral of ln(secx + tanx)?
  1. Derivation of tanx. (Replies: 29)