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I am a high school student and am a beginner in integral calculus. In one of my reference textbooks is said that there were certain integrals which “can't be found”. Some of these include ${\int}{\sin x\over x}\ {\rm dx}$, ${\int}{\cos x\over x}\ {\rm dx}$, ${\int}{1\over \log x}\ {\rm dx}$, ${\int}x\tan x \ {\rm dx}$

I graphed these equations in Desmos and found nothing strange. None of the explanations online made any senese and I failed to understand the following-

1)What about these functions makes them non-integrable?

2)Are there infinitely many functions like this?

3)Why does this happen when the curve is continuous and the area is well defined?

José Carlos Santos
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Sam
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1 Answers1

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Those integrals can be found. You can compute them with any precision that want. The problem is that you cannot express them using only elementary functions. This is a very broad class of functions which include probably any differentiable function that you've ever heard of.

José Carlos Santos
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  • But why only these functions? What limitation of these functions makes it impossible to express using elementary functions? – Sam Sep 26 '19 at 18:01
  • These functions are elementary functions. It's their integrals that cannot be expressed using only elementary functions. And actually the integrals of most elementary functions have that problem. – José Carlos Santos Sep 26 '19 at 18:03
  • I know that these functions are elementary. But what about them makes their integral non-elementary? Why dont we face the same problem while integrating something like $sinx$ or $cosx$? – Sam Sep 26 '19 at 18:17
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    I suggest that you read about Liouville's theorem. – José Carlos Santos Sep 26 '19 at 18:21
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    @Sam: A related question: https://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral – Hans Lundmark Sep 26 '19 at 18:39