1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Difficult Improper Integral Involving Arctan(x)

  1. May 31, 2012 #1
    So recently I've been working through some challenge problems from my old calculus textbook for fun. I'm stuck on one of the integrals, though, and can't find any solutions online. This isn't for homework...it's for my interest and hopefully the interest of others. Here it is (sorry about the formatting):

    int((arctan(Pi*x)-arctan(x))/x, x = 0 .. infinity)

    In the problem statement it says you first need to express the integral as an iterated integral. I also know the answer is (π/2)ln(π).
  2. jcsd
  3. May 31, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Interesting problem ...

    [itex]\displaystyle \int_0^\infty \frac{\arctan(\pi x)-\arctan(x)}{x}\,dx[/itex]

    That this may come from an iterated integral suggests:
    \int_0^\infty \left(\displaystyle \left.\arctan(y)\right|_{y=x}^{y=\pi x}\right)\frac{1}{x}\,dx[/itex]​

    arctan(y) is the anti-derivative of what function?

    If you write this as an iterated integral, what happens if you change the order of integration ?
  4. May 31, 2012 #3
    The answer is zero.

    Actually, because each term:
    \int_{0}^{\infty}{\frac{\arctan x}{x} \, dx}
    is logarithmically divergent on the upper bound, we need to be more careful. Your integral reduces to the following limit:
    \lim_{B \rightarrow \infty} \int_{B}^{B \pi}{\frac{\arctan x}{x} \, dx}
    [strikeout]Make the substitution [itex]x = \tan(z/2)[/itex]. You will get:
    2 \,\int^{2\arctan(B/\pi)}_{2\arctan{B}}{\frac{z}{\sin z} \, dz}, \ B \rightarrow \infty

    Scratch the last subst. Try this [itex]x = B y[/itex]. You get:
    \int_{1}^{\pi}{\frac{\arctan(B \, y)}{y} \, dy}, \ B \rightarrow \infty
    Now, in the above limit the arctangent is [itex]\pi/2[/itex], and the remaining integral is [itex]\ln \pi[/itex]. Combining everything, we get:
    \frac{\pi \, \ln \pi}{2}
    Last edited: May 31, 2012
  5. May 31, 2012 #4
    Awesome! Thank you both. Those were just the thoughts I needed to do it myself.
    Sorry for not posting my initial attempts...they both involved rewriting the integral as an iterated integral and switching the order (I figured out two ways of doing this). The trouble was that each time the integral became more complex. Now it makes sense, though.
  6. Jun 1, 2012 #5
    You might be interested to know that in general, if [itex] \frac{f(x)}{x} [/itex] is integrable on any interval [itex] [a , b] [/itex] for [itex] 0<a<b [/itex] then for all [itex] \alpha [/itex], [itex]\beta > 0[/itex]:

    [tex] \int_{0}^{∞} \frac{f(\alpha x)-f(\beta x)} {x} dx = (A-B)\ln(\frac{\beta} {\alpha}) [/tex]

    where [itex] \displaystyle A = \lim_{x \rightarrow 0^{+}} f(x) [/itex] and [itex] \displaystyle B = \lim_{x \rightarrow ∞} f(x) [/itex]
  7. Jun 1, 2012 #6
    Interesting! I think I see how to prove the generalization.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook