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In my book, a list of integrals have been given which the author states

... such anti-derivatives "cannot be found".

Some of the members of the list are as under:

$\int\dfrac{\sin x}{x} dx$ , $\int\dfrac{1}{\log x} dx$, $\int\sqrt{1 - k^2\sin^2x}dx$, $\int\sqrt{\sin x}dx$, $\int\cos(x^2) dx$, $\int x\tan xdx$ , $\int e^{-x^2}dx$, $\int\dfrac{x^2}{1 +x ^5}dx$, $\int\sqrt{1 + x^3}dx$.

The author only mentions that:

...not every anti-derivatives, even when it exists, is expressed in closed form ....

Now, can anyone tell me why these integrals "cannot be found"? Are they discontinous, non-differentiable or what? Also, what did the author mean by closed form??

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  • $\begingroup$ If it's an anti-derivative, it sure better be differentiable... "Closed form" usually means in this context, "Can be written as a combination of elementary functions, e.g. ones of the form $\sin,\cos,\tan,\exp,\log$, constants, polynomials, etc" $\endgroup$ – Hayden Aug 14 '15 at 14:16
  • $\begingroup$ Are you sure $\int\frac{x^2}{1+x^5}dx$ is on that list? $\endgroup$ – Akiva Weinberger Aug 14 '15 at 14:44
  • $\begingroup$ @columbus8myhw: Oh! Yes sir! It is included & on your saying I've rechecked again & it is included but don't know why it is included. $\endgroup$ – user142971 Aug 14 '15 at 14:53
  • $\begingroup$ Seems like a typo. $\endgroup$ – Akiva Weinberger Aug 14 '15 at 14:54
  • $\begingroup$ @columbus8myhw: Can you tell, why, sir? $\endgroup$ – user142971 Aug 14 '15 at 14:55
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Remember that a function is simply a rule for turning an input into an output. This means that: $$f(x)=\int_0^xe^{-t^2}dt$$ is a perfectly fine function. The fundamental theorem of calculus tells us that $f(x)$ is an antiderivative of $e^{-x^2}$.

However, $f(x)$ cannot be written in closed-form. What that means, in this context, is that $f(x)$ cannot be written as some combination of polynomials, trig functions, exponents, logarithms, additions, subtractions, multiplications, and divisions. (A function that can be written in such a form is usually called an elementary function. $e^{-x^2}$ does not have an elementary antiderivative.)

The proof that $e^{-x^2}$ has no elementary antiderivative is extremely difficult.

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  • $\begingroup$ +1; Okay, sir, this is new to me; I just wanted to ask you what the proof is but your last line nailed my hopes! Still, I would ask you to atleast provide me a source or a link that deduces the proof. Also, can you tell what the antiderivative looks like without any elementary functions?? Thanks in advance:) $\endgroup$ – user142971 Aug 14 '15 at 15:18
  • $\begingroup$ @user36790 See here for a discussion of that particular integral. $\endgroup$ – Akiva Weinberger Aug 14 '15 at 15:26
  • $\begingroup$ @user36790 To prove it, you need to know a branch of math called "differential Galois theory," which I know nothing about. I've heard that you can prove this from Liouville's theorem and the Risch algorithm, but I never found a good introduction to the subject. $\endgroup$ – Akiva Weinberger Aug 14 '15 at 15:28
  • $\begingroup$ Thanks for the resource; since it is beyond my scope now to understand the theorems Lucian linked, I want to ask you how can I detect this integral can't have anti-derivative in closed-form? $\endgroup$ – user142971 Aug 14 '15 at 15:29
  • $\begingroup$ I'm sorry, I don't know a whole lot about this branch of math. I think that the Risch algorithm is a way to see what functions have antiderivatives and what functions don't, but I've never looked into it. $\endgroup$ – Akiva Weinberger Aug 14 '15 at 15:31
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It means that these antiderivatives exist and are perfectly fine functions, but they can not be expressed by other elementary functions (functions that we are already well familiar with). If such anti derivative comes up very often, it is often given a name. Then you can study its properties and use it like any other function. Such functions are often called "special functions", but there is nothing more special about them than about sine or exponential.

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Closed form means that the integral cannot be expressed in standard mathematical functions like trigonometric, exponential or polynomial functions etc.
See the example of $\int {{2^2}^2}^x dx$. Also, you may want to study elliptic integrals.

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    $\begingroup$ +1;You can tell it as a weird coincidence but do you know which book I am talking about?? Integral Calculus by A. Agarwal! $\endgroup$ – user142971 Aug 14 '15 at 15:31
  • $\begingroup$ You are Indian, aren't ye? $\endgroup$ – Aditya Agarwal Aug 14 '15 at 15:32
  • $\begingroup$ Yes, of course! My profile does bear that:) $\endgroup$ – user142971 Aug 15 '15 at 1:57
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    $\begingroup$ It does?? Didn't see. ;) $\endgroup$ – Aditya Agarwal Aug 15 '15 at 4:15
  • $\begingroup$ Really sorry; I don't know why it is not showing my locations but my other accounts do that:| BTW, happy Independence Day:) $\endgroup$ – user142971 Aug 15 '15 at 5:16
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Can anyone tell me why these integrals “cannot be found” ?

For instance, the roots of a $($general$)$ quintic polynomial cannot be found either, i.e., they cannot be expressed in closed form. Basically, what the Abel-Ruffini theorem is for polynomials, Liouville's theorem and the Risch algorithm are for calculus.

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