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May 23rd, 2017, 10:33 AM  #1 
Newbie Joined: May 2017 From: south korea Posts: 7 Thanks: 0  not definitely integrable function example plz
As you know, some continuous functions doesn’t have definite integral. But in spite of the fact that the function does not definitely integral, if the result of the area below the curvature has significance, we alternate it with approximation as through midpoint rule, trapezoidal rule, or Simson’s rule. I wanna collect such functions whose integralapproximations are used as a replacement of definite integral in the field. (so it implies such function is neither Riemann integrable nor Lebesque integrable) I have googled it for many times, however, failed to find any usage. What I got is only about lists of approximation methods listed above. So I decided to ask your aid. Don’t need to write down all function, just let me know name of function or where I can find. Thank you. Last edited by akuraangran; May 23rd, 2017 at 10:57 AM. 
May 23rd, 2017, 04:25 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
If a function is continuous and nonintegrable, at least one of the integral's limits is infinite. The function could be a nonzero constant, for example.

May 24th, 2017, 03:18 AM  #3 
Newbie Joined: May 2017 From: south korea Posts: 7 Thanks: 0 
Thanks for your response, but I am really sorry to say that is a little bit different from what I want to know. Nonzero constant function can yield the Riemannintegral (definite integral), especially in the R^n space with standard topology. Given that integral is defined in closed subset and nonzero constant is continuous in R^n, it need not to be approximated. The function that I want to find might be somewhat like exp(x)/x, ln(x^n1 +c1)/(x^n2 + c2). Although continuous function, in closed set of R, its definite integral does not exist. So, alternative estimate (as listed 3 methods) is used on its behalf. Thanks for your time. Last edited by skipjack; May 24th, 2017 at 08:21 AM. 
May 24th, 2017, 04:44 AM  #4  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902  Quote:
Quote:
Last edited by skipjack; May 24th, 2017 at 08:25 AM.  
May 24th, 2017, 06:46 AM  #5  
Newbie Joined: May 2017 From: south korea Posts: 7 Thanks: 0  Quote:
Then simply, if f = 3, then integration is done with a range 1 to 10, then its definite integral is 3x +c and Riemann value is 27. But the functions I mentioned above as exp(x)/x and ln(x^n1 +c1)/(x^n2 + c2), are obviously continuous in [1,10] respect to x. However, they don't have definiteintegral forms well matched to them. What I search for is some of those kind of functions and the value of integral is quite useful and may need more accurate approximation for it, if possible. Thanks for your comment. Last edited by skipjack; May 24th, 2017 at 08:28 AM.  
May 24th, 2017, 08:47 AM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,481 Thanks: 2041 
They may not have "definite integral forms" in terms of very familiar functions, such as sin(x), but why not allow the use of other functions, such as the gamma function?


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approximation, function, integrable, nonlebesque, nonriemann, plz 
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