Intereting Posts

Expectation of Minimum of $n$ i.i.d. uniform random variables.
Abstract nonsense proof of the cocompleteness of the category of groups
Must a proper curve minus a point be affine?
Existence of function (AC)
Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}dx$
Determining variance from sum of two random correlated variables
Does smashing always increase the connectivity of a space?
What is a Real Number?
Finding supremum of all $\delta > 0$ for the $(\epsilon , \delta)$-definition of $\lim_{x \to 2} x^3 + 3x^2 -x + 1$
All models of $\mathsf{ZFC}$ between $V$ and $V$ are generic extensions of $V$
How many entries in $3\times 3$ matrix with integer entries and determinant equal to $1$ can be even?
Choosing points in fractions of the unit interval
Is $\int_0^\infty\frac{|\cos(x)|}{x+1} dx$ divergent?
PDF of product of variables?
Intuitive Explanation of Morphism Theorem

There are special integrals such as the logarithmic integral and exponential integrals. I want to know if there are primitives for such integrals. If not, why not?

- How to prove these inequalities: $\liminf(a_n + b_n) \leq \liminf(a_n) + \limsup(b_n) \leq \limsup(a_n + b_n)$
- Is there a function that is contiunous at all the rationals, and discontinuous at all the irrationals?
- Number of $\sigma$ -Algebra on the finite set
- Show $\lim\limits_{n\to\infty} \frac{2n^2-3}{3n^ 2+2n-1}=\frac23$ Using Formal Definition of Limit
- How smooth can non-nice associative operations on the reals be?
- $\sum \limits_{n=1}^{\infty}n(\frac{2}{3})^n$ Evalute Sum
- If $p$ is a non-zero real polynomial then the map $x\mapsto \frac{1}{p(x)}$ is uniformly continuous over $\mathbb{R}$
- Uniform convergence of geometric series
- Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
- Uniform convergence for sequences of functions

A simple starting point (as indicated by Qiaochu) is Liouville’s theorem (or ‘principle’) based on differential algebra and was extended with the Risch algorithm.

This last link should clarify some of the ideas used :

- the only new term appearing during an integration (i.e. that was not in the integrand) is a linear combination of logarithms (because logarithms alone may disappear during differentiation…)
- exponentials $e^f$ had to be in the integrand first (since differentiation doesn’t make them disappear) and will reappear as $h\,e^f$ (of course subtle points exist like considering $\sqrt{x}=e^{\,\large{\ln(x)/2}}\cdots$)
- differentiation of an algebraic function $\theta$ (i.e. there is a polynomial $P(\theta)=0$) will give a rational function $\dfrac {d(\theta)}{e(\theta)}$ with $\,d$ and $e\,$ two polynomials.

These 3 ideas will provide logarithmic, exponential and algebraic extensions to the differential algebra (starting for example with the field of rational functions over $\mathbb{Q}$) that will give all the elementary functions.

An excellent tutorial about this is “Symbolic Integration” from Manuel Bronstein.

Geddes, Czapor and Labahn’s book “Algorithms for Computer Algebra” is very clear too.

Now let’s use these ideas to study $\;\displaystyle\int\frac {e^x}x\,dx$.

From a more precise version of $2.$ a primitive must be of type $\ I(x)=h(x)\,e^x\;$ with $h(x)$ a rational function. Let’s suppose this and differentiate $\,I(x)$ :

$$(h'(x)+h(x))\,e^x=\frac {e^x}x$$

so that we need :

$$h'(x)+h(x)=\frac 1x$$

We supposed $h$ rational so that it may be decomposed in simple elements but $h'(x)$ can’t give $\dfrac 1x$ so that $\dfrac 1x$ must be part of $h(x)$. In this case $h'(x)$ will create a term $-\dfrac 1{x^2}$ that must be compensated by a $\dfrac 1{x^2}$ term inside $h(x)$ that will generate a $-\dfrac 2{x^3}$ term… This process clearly doesn’t end !

The same method could be used for the sine integral : $\;\displaystyle\int \frac {\sin(x)}x\,dx\,$ simply by writing $\ \sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$.

(this method was presented by Matthew P Wiener in an old post at sci.math : recommended reading too !)

Concerning the logarithmic integral we have $\ \operatorname{li}(x)=\operatorname{Ei}(\ln(x))\ $ so that the non-elementary proof for the one should apply for the other as well.

- The definition of the logarithm.
- I don't understand how sets can be closed, yet disjoint?
- Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra.
- Convergence of the sequence $\frac{1}{n\sin(n)}$
- How many words can be formed from 'alpha'?
- inverse of diagonal plus sum of rank one matrices
- How to interpret the adjoint?
- Who named “Quotient groups”?
- “+”-Sets are measurable.
- Find $\lim\limits_{(x,y) \to(0,0)} \frac{xy^2}{ x^2 + y^4} $
- A variant of the Knight's tour problem
- Sum of positive definite matrices still positive definite?
- What are the restrictions on the covariance matrix of a nonnegative multivariate distribution.
- Find the Vectorial Equation of the intersection between surfaces $f(x,y) = x^2 + y^2$ and $g(x,y) = xy + 10$
- Fixed Set Property?