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I have been baffled for a while now because no antiderivative has been found for what seems to be a simple enough Integral on the surface.

$$\text{Si}(x)= \int \frac{\sin(x)}{x} dx $$

Is this due to the fact that we do not know what its sum approximates? Namely :

$$\sum_{k=0}^{\infty} \frac {(-1)^k x^{2k+1}}{(2k+1)(2k+1)!} = \text{Si}(x) $$

I have noticed that Si(x) , Ci(x) , Ei(x) all contain sums in their definitions which follow the pattern.

$$ \sum_{k=t }^{\infty}\frac{\text{something}}{a \cdot a!}$$

Do we simply not know what these sums approach or do not know how to tackle the integral form, or is it impossible to give the antiderivative in terms of elementary functions??

Thank you very much for your time and help.

Adam Jamil
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    This is a tough question, because it does have an antiderivative: $\int_1^x \mathrm{Si}(t) \ dt$ (on a suitable interval avoiding $0$). It's just not the form you want. – Randall Jan 08 '19 at 03:10
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    It is indeed impossible to find an antiderivative in terms of elementary functions. See here:https://en.m.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) Also here: https://math.stackexchange.com/q/155/515527 – Zacky Jan 08 '19 at 03:12
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    $Si$ is already not an elementary function. We've essentially just tacked that name onto the antiderivative because we refer to it often. If we cared about the antiderivative of $Si$ enough, it would also have a name. – Matt Samuel Jan 08 '19 at 03:13
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    Please take note that $sinc(x):=\frac{sin(x)}{x}$ itself has been recognized very lately (XXth century) as an important function in itself, and its antiderivative as well : Whittaker said in a funny way that it's "a function of royal blood whose distinguished properties set it apart from its bourgeois brethren". – Jean Marie Jan 08 '19 at 08:10
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    Think about it: does the easy function 1/x have an antiderivative, or do we just call the antiderivative ln? – lalala Jan 08 '19 at 19:17

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