Integral equation

  • #1
Rectifier
Gold Member
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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?
 

Answers and Replies

  • #2
BvU
Science Advisor
Homework Helper
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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.
 
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  • #3
Rectifier
Gold Member
313
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Okay, thank your for your help.
 

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