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!

Challenging Integrals For Fun (1):

  1. Feb 8, 2010 #1
    Hello :)
    Here we go! :


    1. [tex]
    \int \frac{dx}{x^3+x^2+x+1}
    [/tex]

    2. [tex]
    \int_{\frac{\pi}{2}}^{\pi} \left( sin(x) ln(x) - \frac{cos(x)}{x}\right) dx
    [/tex]

    Do your best :)
     
  2. jcsd
  3. Feb 8, 2010 #2
    1 + x + x^2 + x^3 = (1-x^4)/(1-x)

    So, we have to integrate -(x-1)/(x^4 - 1)

    The singularities of this function are at x = -1, i and -i, so the partial fraction decomposition is:

    -1/2 1/(x+1) +1/4 (1+i)/(x-i) + 1/4 (1-i)/(x+i)

    The integral is thus given by:

    -1/2 Log(x+1) + 1/4 (1+i)Log(x-i) + cc of last term. =

    -1/2 Log(x+1) + 1/2 Re[(1+i)Log(x-i)]

    Real and imaginary parts of logarithms of complex arguments are easily obtained as follows. We have:

    Log[r exp(i theta)] = Log(r) + i theta

    This means that:

    Log(x + i y) = 1/2 Log(x^2 + y^2) + i arctan(y/x)

    The integral is thus given by:

    -1/2 Log(x+1) + 1/4 Log(x^2 + 1) + 1/2 arctan(1/x)

    -1/2 Log(x+1) + 1/4 Log(x^2 + 1) - 1/2 arctan(x)

    (pi absorbed in integration constant which we don't write down)
     
    Last edited: Feb 8, 2010
  4. Feb 8, 2010 #3
    Um, I believe #1 is just partial fractions, while #2 is integration by parts. No need to be fancy :)
     
  5. Feb 8, 2010 #4
    sin(x)ln(x)-cos(x)/x=(-cos(x))'ln(x)+(-cos(x))(ln(x))'=(-cos(x)ln(x))'
     
  6. Feb 9, 2010 #5

    berkeman

    User Avatar

    Staff: Mentor

    Welcome to the PF, Ratio. We generally do not allow homework-like brain teasers here on the PF, for obvious reasons. Please do not post this type of question again. I've moved this question to the Homework Help forums, and the normal homework rules apply (we don't do these types of questions for students).
     
  7. Feb 12, 2010 #6

    One Question : Who told you this is for my homework ?
    Did you read the thread's title ?
     
  8. Feb 12, 2010 #7

    berkeman

    User Avatar

    Staff: Mentor

    Don't imply I'm an idiot. If you don't understand why homework questions aren't allowed as brain teasers, please think about it a bit more. And re-read the PF Rules link at the top of the page that you agreed to when you joined here.

    Thread locked. Show some brains people.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Challenging Integrals For Fun (1):
  1. A challenging Integral (Replies: 8)

  2. Fun Integral (Replies: 19)

  3. Challenging integral (Replies: 25)

  4. Integral challenge (Replies: 7)

Loading...