Mixing
List of functions not integrable in elementary terms
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When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary? I'm thinking of things like $exp(x^2)$ which is the standard example, $sin(exp(-x))$ perhaps, things like this, not hugely complicated formulae.
integration reference-request elementary-functions
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
$endgroup$
add a comment |
$begingroup$
When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary? I'm thinking of things like $exp(x^2)$ which is the standard example, $sin(exp(-x))$ perhaps, things like this, not hugely complicated formulae.
integration reference-request elementary-functions
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
$endgroup$
2
$begingroup$
$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
$endgroup$
– Lucian
Feb 18 '14 at 6:00
$begingroup$
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
$endgroup$
– David
Feb 18 '14 at 6:04
$begingroup$
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
$endgroup$
– Martin Sleziak
Jul 4 '14 at 8:04
add a comment |
$begingroup$
When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary? I'm thinking of things like $exp(x^2)$ which is the standard example, $sin(exp(-x))$ perhaps, things like this, not hugely complicated formulae.
integration reference-request elementary-functions
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
$endgroup$
When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the examples I give mostly come from "folklore" or guesswork.
Can anyone point me to a list (not a complete list of course!) of fairly simple elementary functions whose antiderivatives are not elementary? I'm thinking of things like $exp(x^2)$ which is the standard example, $sin(exp(-x))$ perhaps, things like this, not hugely complicated formulae.
integration reference-request elementary-functions
integration reference-request elementary-functions
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
edited May 15 '14 at 8:53
Martin Sleziak
45k10122277
45k10122277
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
asked Feb 18 '14 at 5:55
DavidDavid
69.8k668131
69.8k668131
2
$begingroup$
$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
$endgroup$
– Lucian
Feb 18 '14 at 6:00
$begingroup$
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
$endgroup$
– David
Feb 18 '14 at 6:04
$begingroup$
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
$endgroup$
– Martin Sleziak
Jul 4 '14 at 8:04
add a comment |
2
$begingroup$
$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
$endgroup$
– Lucian
Feb 18 '14 at 6:00
$begingroup$
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
$endgroup$
– David
Feb 18 '14 at 6:04
$begingroup$
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
$endgroup$
– Martin Sleziak
Jul 4 '14 at 8:04
2
2
$begingroup$
$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
$endgroup$
– Lucian
Feb 18 '14 at 6:00
$begingroup$
$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
$endgroup$
– Lucian
Feb 18 '14 at 6:00
$begingroup$
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
$endgroup$
– David
Feb 18 '14 at 6:04
$begingroup$
@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
$endgroup$
– David
Feb 18 '14 at 6:04
$begingroup$
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
$endgroup$
– Martin Sleziak
Jul 4 '14 at 8:04
$begingroup$
It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
$endgroup$
– Martin Sleziak
Jul 4 '14 at 8:04
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
answered Feb 18 '14 at 6:01
sassas
2,49511126
$endgroup$
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
add a comment |
$begingroup$
Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Note:
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
$endgroup$
add a comment |
$begingroup$
The reference below treats as example six different classes of simple nonelementary integrals.
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
answered Jan 31 at 21:15
IV_IV_
1,556525
$endgroup$
add a comment |
$begingroup$
Can you calculate the indefinite integral of this function? I have tried several techniques, (I even tried using Wolfram Alpha) but no luck. I know that it converges but thats all I know. Here is the function though: f(x) = (-x^2)/((x^x)+x)
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
$endgroup$
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
add a comment |
Your Answer
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
answered Feb 18 '14 at 6:01
sassas
2,49511126
$endgroup$
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
add a comment |
$begingroup$
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
answered Feb 18 '14 at 6:01
sassas
2,49511126
$endgroup$
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
add a comment |
$begingroup$
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
answered Feb 18 '14 at 6:01
sassas
2,49511126
$endgroup$
Try this link. A lot of simple functions, btw :)
http://calculus-geometry.hubpages.com/hub/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
As was said in the comment below, the link doesn't work now.
Still, nothing could be deleted from the Internet permanently.
http://web.archive.org/web/20160612175604/http://hubpages.com:80/education/List-of-Functions-You-Cannot-Integrate-No-Antiderivatives
answered Feb 18 '14 at 6:01
sassas
2,49511126
edited Jul 21 '17 at 22:37
answered Feb 18 '14 at 6:01
sassas
2,49511126
answered Feb 18 '14 at 6:01
sassas
2,49511126
answered Feb 18 '14 at 6:01
sassas
2,49511126
2,49511126
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
add a comment |
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
Thanks @sas, exactly what I wanted. Loved the item on "curious exceptions".
$endgroup$
– David
Feb 18 '14 at 6:38
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
$begingroup$
The link does not work anymore.
$endgroup$
– projectilemotion
Jun 13 '17 at 7:35
add a comment |
$begingroup$
Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Note:
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
$endgroup$
add a comment |
$begingroup$
Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Note:
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
$endgroup$
add a comment |
$begingroup$
Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Note:
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
$endgroup$
Liouville's theorem in fact exactly characterizes functions whose antiderivatives can be expressed in terms of elementary functions.
However, the only proof I have seen is not exactly suitable for teaching beginning calculus students. In fact, the proof of the impossibility of solving a general 5th degree polynomial by radicals (by Galois) and the proof of Liouville's theorem share a common idea.
(Liouville's theorem is part of what is called differential Galois theory)
If you are prepared to wade through a bit of differential Galois theory to get to the proof, you could read R.C.Churchill's notes available here.
You could also try Pete Goetz's presentation here
which assumes Liouville's theorem and proves the the Gaussian does not have a elementary antiderivative.
Note:
Proving that a certain function does not have an elementary antiderivative
is often quite difficult, and reduces to the problem of showing that a certain differential equation does not have a solution.
I have not seen many examples of such functions, and I do not know a reference which proves it for all the functions listed in the previous answer by sas.
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
edited Jul 7 '14 at 4:55
Martin Sleziak
45k10122277
45k10122277
answered Jul 6 '14 at 21:35
user56914
add a comment |
add a comment |
$begingroup$
The reference below treats as example six different classes of simple nonelementary integrals.
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
answered Jan 31 at 21:15
IV_IV_
1,556525
$endgroup$
add a comment |
$begingroup$
The reference below treats as example six different classes of simple nonelementary integrals.
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
answered Jan 31 at 21:15
IV_IV_
1,556525
$endgroup$
add a comment |
$begingroup$
The reference below treats as example six different classes of simple nonelementary integrals.
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
answered Jan 31 at 21:15
IV_IV_
1,556525
$endgroup$
The reference below treats as example six different classes of simple nonelementary integrals.
Yadav, D. K.: A Study of Indefinite Nonintegrable Functions. PhD thesis, Vinoba Bhave University, India, 2012
Yadav, D. K.: Six Conjectures in Integral Calculus. 2016
Yadav, D. K.: Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions. 2016
answered Jan 31 at 21:15
IV_IV_
1,556525
edited Feb 2 at 22:34
answered Jan 31 at 21:15
IV_IV_
1,556525
answered Jan 31 at 21:15
IV_IV_
1,556525
answered Jan 31 at 21:15
IV_IV_
1,556525
1,556525
add a comment |
add a comment |
$begingroup$
Can you calculate the indefinite integral of this function? I have tried several techniques, (I even tried using Wolfram Alpha) but no luck. I know that it converges but thats all I know. Here is the function though: f(x) = (-x^2)/((x^x)+x)
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
$endgroup$
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
add a comment |
$begingroup$
Can you calculate the indefinite integral of this function? I have tried several techniques, (I even tried using Wolfram Alpha) but no luck. I know that it converges but thats all I know. Here is the function though: f(x) = (-x^2)/((x^x)+x)
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
$endgroup$
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
add a comment |
$begingroup$
Can you calculate the indefinite integral of this function? I have tried several techniques, (I even tried using Wolfram Alpha) but no luck. I know that it converges but thats all I know. Here is the function though: f(x) = (-x^2)/((x^x)+x)
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
$endgroup$
Can you calculate the indefinite integral of this function? I have tried several techniques, (I even tried using Wolfram Alpha) but no luck. I know that it converges but thats all I know. Here is the function though: f(x) = (-x^2)/((x^x)+x)
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
answered Mar 7 '18 at 18:19
Michael SiskoMichael Sisko
1
1
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
add a comment |
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
$begingroup$
This would really be better suited as a comment; while it does make a small contribution to the post, it doesn't go towards answering the original question.
$endgroup$
– Robert Howard
Mar 7 '18 at 18:38
add a comment |
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2
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$displaystyleint x^{^{tfrac x{ln x}}}dxqquad$ ;-)
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– Lucian
Feb 18 '14 at 6:00
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@Lucian, can we say that's a " "non-elementary" " integral? - note the double quotes... ;-)
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– David
Feb 18 '14 at 6:04
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It would be nice to have some source which not only gives a list of functions, which are not elementary integrable, but also gives some references pointing to proofs that they are not elementary integrable. That's why I have added a bounty. (But if no such answer appears, I will award bounty to the existing answer, so that the bounty rep is not wasted.)
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– Martin Sleziak
Jul 4 '14 at 8:04