Section 3.1 Solvable Integrals
In Calculus I, I went through many techniques and rules for differentiation (chain rule, product rule, quotient rule, etc), but only one major technique for integration (the substitution rule). In this course, I aim to cover the remaining major techniques of integration.
Integration is much more difficult than differentiation. For elementary functions, the derivative rules are a complete set of tools: any elementary function has a derivative which is another elementary function and that derivative can be calculated using the known rules. For integrals, the situation is entirely different. The rules are much less algorithmic; there is much more guess work involved. More importantly, sometimes none of the rules work at all. There are elementary functions \(f(x)\) such that the integral \(\int f(x) dx\) has no solution which is still an elementary function.
Example 3.1.1.
The Guassian distribution (often called the bell curve) is central to statistics. This is the function \(f(x) =
e^{-x^2}\text{.}\) Consider its integral.
\begin{equation*}
\int e^{-x^2} dx
\end{equation*}
This is a very useful integral, as I shall discuss in
Section 8.1. But its integral is unsolvable by elementary function. By the tools introduced in this course, the integral is unsolvable.
I’ve been talking about integral that are solvable by elementary functions. However, the class of solvable integrals is much larger than this if I allow myself to consider non-elementary functions. An integral \(\int f(x) dx\) might not have a elementary antiderivaitve, but it still might have some other kind of antiderivative. In fact, here is a nice proposition that tells me when antiderivatives exist.
Proposition 3.1.2.
Let \(f: [a,b] \rightarrow \RR\) be a continuous function. Then \(f\) has an antiderivative which is defined on the interval \([a,b]\text{.}\)
This is a remarkable result. Continuous is not a very strong condition. It’s much weaker than, say, differentiable. Sharp corners are fine, piecewise functions that line up are fine. Many, many functions are integrable, have antiderivatives.
So these ‘unsolvable’ integrals actually do have solutions. The solutions are new functions which are the antiderivatives of these old functions. The new functions might just not be elementary functions. They might be new functions, with new names, new properties, new behaviour. In fact, integrals without elementary antiderivatives are one of the most significant source of new functions. There is a whole enormous branch of mathematics that studies all these other functions: names them, calculates their values, determines their properties.
This brings up an important and subtle theme in mathematics: what, really, is a solution? To take the example above, does \(\int
e^{-x^2} dx\) have a solution? If, for a solution, you mean an elemetary function, then the answer is no. If, for a solution, you mean any function (possibly an entirely new and unknown function), then the answer is yes. Whether a mathematical problem has an answer can very much depend on the subtleties of how the question is asked and the assumptions about what possibilities exist.
All that said, in this course I will keep to integrals which have elementary solutions. Even restricted to this domain, there are many techniques and many useful functions whose antiderivatives can be calculated.
