Why can't some functions be integrated?

Question

There are some functions (such as xtan(x)dx [without limits]) which can't be integrated, why? How can one understand which function is integrable or not?

Answer 1

Why doesn't every function have an antiderivative?

Finding the indefinite integral of a function $f$, is the same as finding a function $F$ whose derivative is $f$. As explained here, this requires $f$ to be continuous.

If $f$ is discontinuous, then there may be some value $x$ such that $f(x)$ is undefined. When this happens, it means that $F'(x)$ does not exist, i.e. $F$ is not differentiable at $x$.

There are ways around this, however. For example, you can remove discontinuities or restrict the domain of $f$ so that it is continuous. There are increasingly obscure methods which can be used to integrate discontinuous functions, but I don't know much about them.

Stretching the definition of "integration" to its utmost extreme, I wouldn't be surprised if every function can be integrated.

Why can't every antiderivative be expressed in terms of elementary functions?

The integral of $x\tan(x)$ (ignoring discontinuities) is:

$$\frac{i}{2}(\text{Li}_2(-e^{2 i x}) + x (x + 2 i \ln(1 + e^{2 i x}))) + c$$

Where $i$ is the imaginary unit, and $\text{Li}_2$ is the polylogarithm of order 2. This is a special function, and cannot be expressed in terms of elementary functions.

The reason that antiderivatives cannot always be expressed in terms of elementary functions is that the set of elementary functions is not closed under limits in general. The specific fact that the integral of an elementary function is not always an elementary function is known as Liouville's Theorem.

Answer 2

Let's make a small distinction here between a function being integrable (the area under the curve exists), and a function being expressible in terms of elementary functions. I will describe both situations.

Riemann Integrable:

View this under the lens of analysis, where we say that a function $f(x)$ is Riemann integrable on an interval $[a, b]$ if we can approximate the area under its graph by slicing the interval into subintervals, and summing the areas of the resulting rectangles which sit above and below the function.

We call these approximations the upper and lower sums, and if these sums get closer and closer to the same value, as we make our partitions finer and finer, then we say that function is Riemann integrable.

One of the most fundamental examples we can think of which is not (Riemann) integrable is the Dirichlet function:

$$ f(x) = \begin{cases} 1, \text{If } x \in \mathbb{Q} \\ 0, \text{If } x \notin \mathbb{Q} \end{cases} $$

What happens if we try to form our upper and lower sums? Well, every subinterval, now matter how small we try to make it, will contain both rational and irrational numbers, as both sets are said to be dense in $\mathbb{R}$. So, the supremum on any subinterval will be 1 (there is always a rational number there!), and the infimum on any subinterval will be 0 (there is always an irrational there). Thus, every upper sum is equal to the length of that interval (all rectangles have height 1), and every lower sum is 0. So no matter how we try to cut up the interval, our sums never converge. So it is not Riemann integrable because it is too jumpy.

Non-Elementary Functions:

Then there are functions whose anti-derivative cannot be expressed by a combination of elementary functions, which include: constants, powers of $x$, exponentials $e^{x}$, logarithms $\ln(x)$, trigonometric functions $\sin(x), \cos(x), \tan(x)...$, and arithmetic operations. Whilst many functions can be integrated, there are lots which have a non-elementary anti-derivative, where we cannot express it in terms of these elementary building blocks. That is, while a function may be integrable, that doesn't mean that its anti-derivable lives in the symbolic world.

Common examples are:

• $ \int e^{-x^{2}} dx $ No elementary anti-derivative, instead we have to define a new special function called erf, defined as $\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^{x}_{0} e^{-t^{2}} dt$
• $\int \frac{\sin(x)}{x}dx$ No elementary anti -derivative, giving rise to $\text{Si}(x) = \int^{x}_{0} \frac{\sin(t)}{t} dt$
• $\int \sqrt{1 + x^{3}} dx$ Is not expressible with elementary function, which gives rise the idea of elliptic integrals!

Further reading recommendation: Take a look at the idea of the Risch algorithm, which helps us to integrate elementary functions. The idea is, often in research, we may end up with an integral we need to solve. We try to apply the rules we know, such as substitution, partial fractions, integration by parts, and maybe consulting a big book of integrals. If these fail, we have to decide if there is an expression of the anti-derivative in terms of elementary functions. As it runs out, the Risch algorithm allows us to prove whether a function has an elementary function as an indefinite integral, determining the indefinite integral in the case it exists. It may help you to understand what's going on clearer!