List of functions not integrable in elementary terms

Question

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.

Answer 1

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

Answer 2

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.

Answer 3

This answer is an effort to collate all known non-elementary integrals into a community wiki post so that it can be dynamically supplemented and so that links to proofs can also be included. I'll include some of the other integrals mentioned by the other answers. This also serves the purpose of providing a backup in case the relevant lists go down.

For simplicity, let $T(x)=\sin x,\cos x,\tan x$ be any trigonometric function and likewise $T^{-1}(x)=\sin^{-1} x,\cos^{-1} x,\tan^{-1} x$. The list is ordered such that all functions including $\ln$ are in section $2$ or beyond, all functions including $e^x$ in section $3$ or beyond, and so on.

Abbreviations:

• SLT: Strong Liouville theorem.

0. General cases

• By integration by substitution, if $f(x)$ has a nonelementary antiderivative then so too does $f(\lambda x)$, where $\lambda \in \mathbb{R},\lambda\ne0$.

1. Algebraic type functions

• $p(x)^{1/n}$ for any $n\in\mathbb{Z},n>2$ where $p$ is a polynomial of degree greater than $2$, except when $p$ is a power of a polynomial or when the radical can be simplified. This includes $\sqrt{p(x)}$, where $p$ is a polynomial subject to the relevant constraints. (1)

• Some Lamé curves, $L(x) = (1 ± x^m)^{1/n}$, where $m$ and $n$ are numbers (of what set?) greater than $1$, and not $m= n = 2$. This includes $(1 - x^3)^{1/3}$, $(1 - x^4)^{1/2}$, $(1 + x^2)^{1/4}$ and $(1 + x^4)^{1/2}$. See also Wikipedia: Elliptic integral. (1, 4, 7 p. 6)

• $(1 ± x^m)^{-1/n}$ where $m$ and $n$ are numbers (in what set?) greater than $1$, except the case where $m = n = 2$. (1)

• $x^p (a+bx^r)^q$, where none of $\frac{p+1}{r},q$, nor $\left(\frac{p+1}{r}+q\right)$ are integers (proof: Chebyshev's theorem). This includes $\sqrt{1+x^3}$ and $\sqrt{1+x^{-4}}$. (3, 7 p. 10)

• $\displaystyle \frac{x}{\sqrt{1-x^3}}$. (7 p. 6)

2a. Logarithm type functions

• $\displaystyle \frac{p(x)}{\ln x}$, where $p$ is a nonzero polynomial. This includes $\displaystyle \frac{x}{\ln x}$ and $\displaystyle \frac{1}{\ln x}$ (proof: SLT). See also Wikipedia: Logarithmic Integral. (1, 5, 7 pp. 6, 10)

• $\displaystyle \frac{\ln x}{p(x)}$, where $p$ is a non-constant polynomial and not a monomial such as $x^n$. This includes $\displaystyle \frac{\ln x}{x\pm 1}$. (1)

• $\ln \left(p(x)+q(x)\right)$, where $p,q$ are non-constant polynomials and not powers of the same linear function, e.g., $p(x) = (2x-5)^3$ and $q(x) = (2x-5)^7$. (1)

• $\displaystyle \frac{\ln p(x)}{\ln q(x)}$, where $p,q$ are non-constant polynomials and not powers of the same polynomial, e.g., $p(x) = (x^2 + 3)^2$ and $q(x) = (x^2 + 3)^9$. (1)

• $\displaystyle \frac{1}{\ln x + x}$. (1)

• $\ln \ln x$. (1, 7 p. 6)

• $\sqrt{1 ± \ln(x)}$ and $\sqrt{x ± \ln(x)}$. (1)

• $\displaystyle \frac{1}{\ln^2x-x^2}$. (7 p. 6)

2b. Exponential type functions

• $\displaystyle e^{\pm x^n}$ for all $n\in\mathbb{Z},n\ne 0,1$. This includes $\displaystyle e^{\pm x^2}$ (proof: SLT, proof sketch: (8)) and $\displaystyle e^{\pm x^3}$. See also Wikipedia: Error Function. (1, 3, 7 pp. 3, 10)

• $e^{x^{\alpha}}$ for $\alpha\ge 2$. This includes $\displaystyle e^{\pm x^2}$. (2, 6)

• $\displaystyle e^{\pm e^{x}}$. (1)

• $\displaystyle \frac{e^{x}}{p(x)}$ where $p$ is "a polynomial" (presumably nonzero or of degree $1$ or higher?). This includes $\displaystyle \frac{e^{x}}{x}$ (proof: SLT, proof sketch: (8)) and $\displaystyle \frac{e^{x}}{x^2}$. (1, 3, 7 pp.3, 10)

• $x^\alpha e^x$, where $\alpha\notin\mathbb{Z}^+_0$. This includes $\sqrt{x}e^x$. (1)

• $\displaystyle \frac{p(x)+e^x}{q(x)+e^x}$, where $p,q$ "are polynomials" (presumably nonzero but of what degree?) and where $p$ is not the derivative of $q$. This includes $\displaystyle \frac{x}{1+e^x}$ and $\displaystyle \frac{1}{x+e^x}$. (1)

• $\displaystyle e^{p(x)/q(x)}$, where $p,q$ are nonzero polynomials (of a certain degree? such that $q$ does not divide $p$?). (1)

• $\sqrt{x ± e^x}$. (1)

3a. Exponential/Logarithm type functions

• $\displaystyle e^x\ln x$ and $xe^x\ln x$. (1)

• $\displaystyle \frac{e^x}{\ln x}$ and $\displaystyle \frac{\ln x}{e^x}$. (1)

• $\displaystyle e^{\sqrt{\ln x}}$. (1)

• $x^x=e^{x\ln x}$. Proof sketch: (8). (3)

• $\ln(p(x) ± e^x)$, where $p(x)$ is a nonzero polynomial. This includes $\ln(1 ± e^x)$, where we have $\deg(p)=0$. (1)

• $xe^{x^3/3}$. (7 p. 6)

3b. Trigonometric type functions

• $T(x^\alpha)$, where $\alpha$ is a constant but $\alpha\ne 0$ and $\frac{1}{\alpha}$ is not an integer. This includes $\sin (x^2)$ and $\cos(x^2)$. See also Wikipedia: Fresnel integral. Exceptions to this rule that have elementary integrals include $\sin(\sqrt{x})$, as $\frac1\alpha=2$. (1, 7 p. 6)

• $x^\alpha T(x)$, where $\alpha \ne 0$, except for when $T(x)=\sin x,\cos x$ and $\alpha \in\mathbb{Z}^+_0$. This includes $\sqrt{x}\sin x$, $\sqrt{x}\cos x$, $x \tan x$ and $\displaystyle \frac{T(x)}{x}$, such as $\displaystyle \frac{\sin x}{x}$. See also Wikipedia: Sine integral and Cosine integral. Proof: SLT, proof sketch: (8). (1, 3, 5, 7 p.3)

• $\displaystyle \frac{x}{T(x)}$. (1)

• $\displaystyle \frac{1}{T(x) + x}$. (1)

• $\ln T(x)$, $e^{T(x)}$, and $T(e^x)$. (1)

• $T_1(T_2(x))$, where $T_1(x), T_2(x)=\sin x, \cos x,\tan x$. This includes $\sin(\sin x)$, $\cos(\tan x)$, etc.. (1)

Let $S(x)=\sin x, \cos x$, i.e., $S(x)$ is a trigonometric function that is explicitly not $\tan x$.

• $\sqrt{S(x)}$ (proof: Chebyshev's theorem). In fact $\sqrt{\tan x}$ has an elementary integral. (1, 7 p. 11)

• $\sqrt{S(x)+\lambda}$, where $\lambda\ne 1$ is a constant. (1)

• $\sqrt{1+x^2}S(x)$. (7 p. 6)

Individual cases.

• $\displaystyle \frac{\sqrt{\sin x}}{x}$. (3)

• $\cosh(x^{\alpha})$, where $\alpha\ge 2$. (6)

• $\tan{\sqrt x}$. (1)

• $x^2 \cos( x^2)$. (7 p. 6)

• $\sqrt{2-\sin^2x}$. (7 p. 6)

• $\sqrt{1-k^2\sin^2x}$. (7 p. 6)

3c. Inverse trigonometric type functions

• $\displaystyle \frac1{T^{-1}(x)}$. Proof sketch: (8). (1)

• $\displaystyle \frac{T^{-1}(x)}{x}$ and $\displaystyle \frac{x}{T^{-1}(x)}$. (1)

• $T^{-1}(e^x)$ and $T^{-1}(\ln x)$. (1)

• $\displaystyle e^{\sin^{-1}\ln x}$ and $\displaystyle e^{\cos^{-1}\ln x}$. (1)

• $\displaystyle e^{\tan^{-1} x}$. (1)

• $T^{-1}(x^2)$, where $T^{-1}\ne \tan^{-1} x$. (1)

References and abbreviations

1. Hubpages - List of Functions Without Antiderivatives.

2. UCR - Functions with nonelementary indefinite integrals.

3. Sosmath

4. Mathworld

5. Nijimbere - Evaluation Of Some Non-Elementary Integrals Involving Sine, Cosine, Exponential And Logarithmic Integrals: Part I.

6. Nijimbere - Evaluation of the non-elementary integral $\int e^{λx^α} dx$,$α≥2$, and other related integrals.

7. Dharmendra Kumar Yadav - A Study on Nonelementary Functions

8. MSE question 265780.

Answer 4

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