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Can't solve this difficult definite integral for the life of me

  1. Jan 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Definite integral:

    [tex] \int \frac{\ln(x+1)}{x^2+1} dx [/tex] from x=0 to x=1 (sorry I don't know how to do integral boundaries with tex)




    3. The attempt at a solution
    I just am clueless on how to do this, I'm almost 100% sure you are not supposed to find the anti-derivative of the integral and then calculate the definite integral, but need a clever trick
    to calculate the definite integral.

    So the standard tricks like u-substition and partial integration do not work.


    Any tips?
    Thanks!
     
  2. jcsd
  3. Jan 11, 2013 #2

    LCKurtz

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    It isn't an elementary integral. Maple gives an answer in terms of dilog functions and a Catalan number.
     
  4. Jan 11, 2013 #3

    HallsofIvy

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    I would be inclined to use a numerical integration.
     
  5. Jan 11, 2013 #4
    It has an algebraic solution and must be solved algebraically(so no numerical methods). I already said I thought it was not an elementary integral , and said that this one has to be solved in a clever way. Have been stuck on this one for almost a week. The answer is (pi/8)ln(2)
     
  6. Jan 11, 2013 #5
    See my reply, sorry forgot to quote.
     
  7. Jan 11, 2013 #6

    SammyS

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    Try a u-substitution that takes x=0 to u=1 and x=1 to u=0 .

    Possibilities:

    u = 1-t (but I think this won't work.)

    [STRIKE]u = 1/(x+1) - 1 → u+1 = 1/(x+1)[/STRIKE]

    [STRIKE]u = 1/(x-1) + 1 → u-1 = 1/(x-1)[/STRIKE]

    Added in Edit:

    (Those last two aren't as advertized. DUH !)

    Try [itex]\displaystyle \ \ u=\frac{1-x}{1+x}\ .[/itex]
     
    Last edited: Jan 11, 2013
  8. Jan 11, 2013 #7

    I got stuck using that substitution, it just got very messy
     
  9. Jan 11, 2013 #8

    haruspex

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    If you integrate by parts in the obvious way, the first term gives, intriguingly, exactly twice the desired answer. The second term is -∫atan(x)/(1+x).dx.
    By a series of substitutions (x = tan(θ), θ→θ/2, θ→θ-π/4; amounting, I think, to x = tan(θ/2-π/8)) I turned that into some multiple of [itex]\int_{\pi/4}^{3 \pi/4}\frac{\left(\theta-\pi/4\right)}{sin(\theta)}.d\theta[/itex]. Splitting the range around π/2 and combining the θ term with its π-θ partner turned that into integrating cosec. Haven't checked it produces the right answer.
     
  10. Jan 11, 2013 #9
    I notice you have 1/(x^2+1). Try a u-substitution where u= 1/(x^2+1).
     
  11. Jan 11, 2013 #10

    SammyS

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    For those of you helping out, as LCKurtz noted in post #2, the indefinite integral cannot be expressed in terms of elementary functions. However, the definite integral can be evaluated using a "trick".

    The result of the substitution [itex]\displaystyle \ \ u=\frac{1-x}{1+x}\,,\ [/itex] is fairly clean.

    [itex]\displaystyle \ \ u=\frac{1-x}{1+x}=\frac{2}{1+x}-1[/itex]

    So that [itex]\displaystyle \ \ u+1=\frac{2}{1+x}\ .\ \ [/itex] This substitution behaves quite nicely.

    This gives [itex]\displaystyle \ \ x+1=\frac{2}{1+u}\,,\ \ [/itex] so finding dx is easy.

    Also note that [itex]\displaystyle \ \ 1+x^2=1+\left(\frac{1-u}{1+u}\right)^2=2\frac{1+u^2}{(1+u)^2}\ .[/itex] Therefore, lots of things cancel out.
     
  12. Jan 11, 2013 #11
    Ok, I shall try again.

    [tex] dx = \frac{-2}{(1+u)^2} du [/tex]

    So the integral becomes

    [tex]\int_{1}^{0} \frac{\ln(\frac{2}{1+u})}{\frac{2(1+u^2}{(1+u)^2}} \frac{-2}{(1+u)²} du = -\int_{1}^{0} \frac{\ln(\frac{2}{1+u})}{1+u^2} du = \int_{0}^{1} \frac{\ln(\frac{2}{1+u})}{1+u^2} du [/tex]

    I don't see how I can get the definite integral with this substitution :-(
     
  13. Jan 11, 2013 #12

    haruspex

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    Try to simplify ln(2/(1+u)), and compare it with the integral you started with.
     
  14. Jan 11, 2013 #13

    Dick

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    You are practically there. Use the rules of logs to break the integral into two parts. Notice something about one of them.
     
  15. Jan 11, 2013 #14
    Oh Wow, got it now. How does one come up with this substitution?
     
  16. Jan 11, 2013 #15
    Thanks ! I got it now, how do you come up with a substitution like this?
     
  17. Jan 12, 2013 #16

    haruspex

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    A frequent trick with nasty definite integrals is to find a way to fold the range so that the nastiness in one part cancels the nastiness in the rest. Or, what is really the same thing, reverse the whole range and reform it so that the original integral pops out as only part of the expression, and with a factor other than 1 (in this case, -1).
    Since the ln(x+1) was the nasty part, that's the part to study for working out the point around which to pivot the range. To make it easier, you could first write x-1 wherever there's an x, so we get ln(x), and the range becomes 1 to 2. A natural way to pivot the range given ln(x) is to map x to 1/x, so ln(x) becomes -ln(x).
    The present problem generalises to ##\int_a^b\frac{ln(x)dx}{x^2+Bx+ab}##
     
  18. Jan 12, 2013 #17
    Okay, thanks!

    But is there a way to see beforehand what would maybe work and what wouldn't ? or do you just have to try?
     
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