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Integral equation

  1. Jan 5, 2017 #1

    Rectifier

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    The problem
    I want to find ##x## which solves ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt = 0 ##

    The attempt
    ##\int^x_1 \frac{\sin t}{t} \ dt = x -1 ## I see that the answer is ##x=1## but I want to be able to calculate it mechanically in case if I get similar problem with other elements. Any suggestions on how I can do that?
     
  2. jcsd
  3. Jan 5, 2017 #2

    BvU

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    Hi,
    'mechanically' sounds good. But, ##{\sin x\over x}## is (and I https://owlcation.com/stem/How-to-Integrate-sinxx-and-cosxx [Broken])

    one of the simplest examples of non-integrable functions in the sense that their antiderivatives cannot be expressed in terms of elementary functions, in other words, they don't have closed-form antiderivatives.​

    However, apart from ##x=1## there shouldn't be too many other solutions ... ##x-1## grows faster than the integral.
    You could also investigate domain [0,x] : with ##{\sin x\over x} < 1## the integral is always different from x-1.
     
    Last edited by a moderator: May 8, 2017
  4. Jan 5, 2017 #3

    Rectifier

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    Okay, thank your for your help.
     
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