Validity of Problem Solving Methods

I was in class the other day, and I was suddenly concerned with the idea that some methods don't work for solving certain problems. I'm working on integration right now, so some of the problem solving methods that I'm talking about involve integration by parts, u-substitution, trigonometric substitution, among others.

The fact that we hit road blocks if we tackle a certain problem with a certain method seems strange to me. If there is a solution to the problem, and granted your methods aren't so off track as though we were trying to integrate by differentiating or something, or in other words that you are using a method that fits the type of problem, why shouldn't all of them work? I can see that some of them will save time, but certainly all should be able to solve them eventually.

If the solution is there, shouldn't the mathematics be able to reveal it by any of its appropriate means?

Any advice on tagging is appreciated.

• Knowing the right approach comes with practice, experience and creativity. You might know many ways of traveling from India to USA, but if I restrict you to traveling only on foot, you are gonna have a bad time. – Calvin Lin Feb 26 '14 at 6:48

Think of integration methods as individual single-function tools in a toolbox. Sometimes there's an ideal tool for a specific fix-it job, and you may even be able to use more than one tool for a job, but you can't get by with one tool and expect to fix everything that breaks. Nor can you expect to fix everything that breaks with any number of tools.

Compare your question to taking derivatives (which comparatively, is a far easier task in general). Certainly the power rule cannot be used in place of the chain rule. So why would you expect a single method for integration to always work?

With different classes of functions we need different methods. And for some functions, there is no easy answer. For example,

$$\int e^{x^2} dx$$

has no "nice" answer. In fact a function, called $\text{erfi(x)}$ was simply defined to be a certain constant times this integral. For many seemingly nice functions, like this one, its Power Series is often substituted instead.

Welcome to mathematics (in fact, welcome to life). Some types of problems have algorithms: these are guaranteed to work all the time. Others do not.

As a matter of fact, there is an algorithm for indefinite integration in elementary functions: the Risch algorithm. However, this is much too complicated for computation by hand except in a few simple cases.

• Am I incorrect in my assumption that the "road blocks" I mentioned truly are insurmountable, or is it just that it simply would take a lot longer and would be a lot more complicated to continue solving the problem with the given method such that I should abandon that path for a better one? – Adam Feb 26 '14 at 7:05
• Some of the "road blocks" really are insurmountable. – Robert Israel Feb 26 '14 at 15:41

Up to a certain point in your mathematical education you basically only get shown problems that can be solved by the methods you're being taught. However, maybe, eventually you realise these methods often only provide small advances into the great intractable jungle of problems. Realising the scale of how much you don't know is the beginning of wisdom (so I've been told), so congratulations on your disillusionment.