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I've been messing around with indefinite integrals. Watching some youtube videos and I found the Sinc function and that it has no finite Anti-derivative. Desmos being my favourite program ever I decided to graph to the equations $y=\frac{\sin(x)}{x}$ and $y=\int_0^x\frac{\sin(a)}{a}da$ (only way I could do indefinite integrals in Desmos)

If you do this you might notice $y=\frac{\sin(x)}{x}$ looks very similar to $y =\sinh(x)$ and $y=\int_0^x\frac{\sin(a)}{a}da$ looks very similar to $y=\arctan(x)$

I wondered how we know that $\frac{\sin(x)}{x}$ has no finite anti-derivative because these simple functions give seemingly accurate approximations

Best approximation I could get for it in the short time I had was: $$\int \sinh(x)-\frac{d}{dx}\bigg[\frac{\sin(x)}{x}\bigg]-\frac{\pi}{2}$$ or $$2\arctan(e^x)+\frac{\sin(x)}{x^2}-\frac{\cos(x)}{x}-\frac{\pi}{2}$$ (In all the examples I'm assume $C=0$ because it's not important and I'm assuming any points with $\frac{0}{0}$ evaluate to their limits as $x$ tends to $0$)

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marked as duplicate by Hans Lundmark, joriki, zipirovich, Lord Shark the Unknown, Key Flex Aug 20 '18 at 0:58

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    $\begingroup$ It has an antiderivative, in fact $\int_0^x \frac{\sin(y)}{y} dy$ is a special function called the sine integral function. It is not an elementary function however. $\endgroup$ – Ian Aug 19 '18 at 14:59
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    $\begingroup$ I think what you wanted to ask is, why is it that the integral of $\frac{\sin x}{x}$ cannot be expressed in terms of so-called elementary functions? to which I would retort, Why do you expect that it should, given that in some sense there are many more functions than the ones that are known or frequently used, or well studied. $\endgroup$ – Allawonder Aug 19 '18 at 15:12
  • $\begingroup$ Mathematics education tends to spend a lot of time on the "pretty" examples. Things that work out nice. It tends to give students the impression that mathematical things tend to work out nice. They don't. $\endgroup$ – steven gregory Aug 19 '18 at 15:20
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    $\begingroup$ I am sure there is a specific proof that $\int \frac {\sin x}{x}dx$ cannot be expressed in term of elementary functions. But when you look at what the precise meaning of "in terms of" is, you will see that this kind of problem can be complicated and deep. $\endgroup$ – DanielWainfleet Aug 19 '18 at 15:36
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    $\begingroup$ @DanielWainfleet: The accepted answer to How to determine with certainty that a function has no elementary antiderivative? includes such a proof. $\endgroup$ – Dave L. Renfro Aug 19 '18 at 16:29