Why are we able to integrate elementary functions, but not more sophisticated ones? Is the integral not simply the area under the function?

Question

I've gone through classes on differential and integral calculus, but I feel we've mostly just learned different methods for computing integrals, and different questions regarding the computation of derivatives and integrals.

I'd like to know: why exactly is it that these computations work so well for elementary functions like $f(x) = x^k$ or $g(x) = ax + b$, not for other functions? For instance, the different combinations of $f(x)$ and $g(x)$ already begin to cause some trouble, even if they might eventually be integrable, at a first glance this is not immediately obvious for $f(x)^{g(x)}$, or if it is, maybe not for $f(x)^{{g(x)}^{f(x)}}$ or some more complicated functions.

In an attempt to better understand the integrals of simple functions, I want to understand: if we know the integral is nothing else than the area under a function, and we have a perfect way for finding the area under a function (Riemann sums), why is the any problem if we're given a function and we can compute that function, even if we at first might not know the corresponding area? We have the value of the function, i.e. of $y$, at every $x$, so the area should come easily enough, no?

I can look at a graph, and instantly I can say: well, the following other curve could be an integral of the above one.

Ultimately, I'd like to deepen my understanding of calculus, so if someone has recommendation of books or of resources I might look at, I would highly appreciate it. I know this is a vague question, but at the moment I look at a graph of a function and its derivative, and fail to see the relationship between them in a way that could be generalized, or I don't see the bigger picture, so I'm not sure exactly how to frame this properly.

Answer 1

The primary method you probably have for computing integrals is applying the fundamental theorem of calculus, which turns the problem of calculating an integral into the problem of finding an anti-derivative and calculating its values at two points.

The problem you're adressing is two-fold:

1) We don't really like most numbers.

What I mean here is that the "typical" real number is some transcendental thing whose exact value is unlikely to make you happy. In this case, you probably only really care about some finite part of the decimal expansion of the given number, in which case you would calculate the integral exactly the way you propose: Have some approximation scheme for the values of the function and simply compute a suitably large Riemann sum. This is the fundamental idea of numeric integration.

In this vein, it is worth noting that polynomials have nice algebraic properties: If you know the input, you know the universe of the output. Say my polynomial has rational coefficients and I compute it at some $x$. Then I know the result must have the form $\sum_{i=1} a_i x^k$ for $a_i\in \mathbb{Q}$ which means that if I know $x$, I can actually say something about this number for accurately. This is one reason integration of polynomials works out nicely.

2) What does an anti-derivative even look like for a general function?

The other thing going on here is that, when applying the fundamental theorem of calculus, you need to have some idea of what an anti-derivative your function is without actually calculating the anti-derivative as an integral - in that case, you'd have already calculated the integral, which would leave the exercise pointless. Elementary functions have the nice property that their anti-derivatives are also elementary functions and this helps a lot.

You'll notice that elementary functions tend to be solutions to nice differential equations, i.e. $y^{(n)}\equiv 0$ for polynomials of degree at most $n$, $y'=y$ in the case of $\exp(x)$ and $y''=-y$ in the case of $\sin$ and $\cos$, and you can deduce their antiderivatives simply from looking at the equations. Then, you can begin to start studying those equations separately and learn more about them.

$\pi$ and $e$ are not natural constants in any other sense than this: They popped up in problems we wanted to solve, and so we decided that they were important. In a completely parallel fashion, elementary functions become elementary because they arise as solutions to nice problems (or at least pop up in nice problems) and then, we start to work with them.

Answer 2

We can compute definite integrals of more sophisticated functions: they are the definite integral of the function over the interval. What we can't do is simplify that value. If you look at the area under $e^{-x^2}$ between 0 and 1, it has a very simple value that is the physical area of the area under the graph.

A good thing to think about is, of you'd just learned about irrationals, "why can't you find the square root of 2"? And, of course, you can find it. It's just that at some point the procedures that we have for finding it just can't come out nicely in terms of other processes we already know (in the case of irrationals that's ratios of integers and in the case of integrals it's composing the functions and values we think we know--that is a combination of additions, cosines, exponentiations and so forth applied to rationals and nth roots and $\pi$ and so forth). $\int_0^1 e^{-x^2} \ \text{d}x$ can be found, it's a number exactly defined by the area under the curve on the interval. We just can't describe that number in terms of the tools we already have to describe numbers. But integration is really just a new way to describe this number. That's why in many fields of math, and integral is thought of as a closed form. But the evaluation we do in calculus is about simplifying that number. Just like cosine can be simplified for 45 degrees in terms of square roots but can't be simplified at all for $e$ degrees, some integrals just happen to simplify to numbers we know.

Why is it that these computations tend to work out better for simple functions? Most of the time in practice, when an integral works out well it has an antiderivative that is a relatively simple function. There are counterexamples, say the dirchelet integral, but I would argue this still makes up the vast majority of integrals we can actually solve. For this large class, by the fundamental theorem of calculus, then, the integrands that we can evaluate for most of the integrals we can evaluate are derivatives of simple functions. But the way that the limits work out with the rules for derivatives is that they tend to keep simple functions simple. That is, because of the chain rule and power rule and product rule and so forth, the derivatives of an elementary function stay elementary, at least most of the time. If you expand your range a little bit and consider some more advanced techniques we have--like Feynman's trick, or even like the residue theorem--they still tend to give evaluations of elementary functions for elementary functions because of the way the limits work out for partial derivatives or the residue for a simple pole and so forth. So, even though there are many simple integrals we can't simplify, the way the analysis of the real numbers and the kinds of functions we feel comfortable dealing with work out is that almost every integral we can simplify is pretty elementary. And if course when we do math in real life we work with things that we like to deal with whenever we can so we adapt underlying models and ideas and so forth that make elementary functions what we have to work with most often when doing integrals. So, both for simplicity and later for preparedness, in calculus classes especially we like to show people things that deal with elementary functions. The intuition behind the calculus sometimes unfortunately gets obscured, but it's a very useful thing, and especially if an integral is just one part of a real problem then it's much nicer to work with closed forms. The result of all of this is that we end up teaching people very computational calculus, but that doesn't change the fact that the calculus isn't that computation. It's the understanding of the integral as the area. Often it also happens that that's enough in practice because of Riemann sums, because we can get a great approximation of the number, but if you can keep your mind around the fact that the essence of what's going on here isn't the simplified value or the approximation, it's the literal area that's drawn and we just can't describe very nicely besides by saying it is that area.