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I Closed form solution for this integral

  1. Feb 14, 2017 #1
    After series of algebraic simplifications, I ended up with the following integral:

    ##\int_0^\infty \exp(-Kx) \arctan(x) dx ##

    As far as I searched, there is no closed form solution for the integral. But, K is my design variable that I need to optimize later. To do this, I need to take K out of the integral. Can you manipulate this integral to express e.g., as a closed form function of K, multiplied with an integral independent of K?
     
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  3. Feb 14, 2017 #2

    blue_leaf77

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    I guess there is a closed form in terms of cosine and sine integrals, Ci(x) and Si(x) respectively. If you are interested in this way, try solving that integral using partial integration first by choosing ##dv = e^{-Kx}dx## and ##u = \arctan x##. For the integral ##\int v \ du## you will get
    $$
    \int_0^{\infty} \frac{\exp(-Kx)}{1+x^2} \ dx
    $$
    This integral can be shown to have the result expressible in terms of ##\textrm{Ci}(K)## and ##\textrm{Si}(K)##. These integrals have a "friendly" derivative when you want to find the optimal ##K##. Although the ultimate solution for ##K## might not be solvable analytically.
     
  4. Feb 15, 2017 #3
    Thank you blue_leaf77. I tried but it is still very difficult to solve for optimal K. My overall expression for the objective is in fact a function of this integral. Here I only provided the most problematic part.

    After integration by parts I have,

    ## \begin{aligned}
    \frac{1}{K}\int_0^\infty \frac{\exp(-Kx)}{1+x^2}dx &= \frac{i}{2K}\left[ e^{iK}\text{Ei}(-K(x+i)) - e^{-iK}\text{Ei}(-K(x-i)) \right]^{x=\infty}_{x=0}\\
    &=\frac{i}{2K}\left[ -e^{iK}\text{Ei}(-iK)) + e^{-iK}\text{Ei}(iK) \right]\end{aligned}##

    where ##\text{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}{t}dt## or ##\text{Ei}(x)=-\int_{1}^\infty\frac{e^{tx}}{t}dt##. This is the same as what you said when ##\text{Ci(x)}## and ##\text{Si(x)}## are replaced with ##\text{Ei(x)}##.
     
  5. Feb 15, 2017 #4

    blue_leaf77

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    If you can get the derivative of ##\textrm{Ei}(x)## in terms of basic functions I think that will be helpful, have you found it?
     
  6. Feb 15, 2017 #5
    The derivative of ##\text{Ei}(x)## can be calculated, but the overall optimization problem I am attempting cannot be solved simply by setting the derivative to zero.
     
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