Introduction: Mental Foundations

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Introduction: Mental Foundations

Mental FoundationsThis short book teaches how to evaluate many integrals mentally. For the most part, if you can now evaluate an integral by writing it out on paper, you can potentially learn to solve it just through thought. With relatively complex integrations, in which you would normally need to write out many steps to solve, you will be able to solve by writing just a few steps down.

You won't learn here to solve every integral that exists; many real-world integrals can only be evaluated through software, and the methods here will not solve those. Fortunately, what you learn will apply in some way to essentially every integral you come across. Even if it is the kind that must be evaluated with a computer, these methods will help you work with them more easily, with greater intuition and understanding.

When you are learning to solve some math that is new to you, there is a process that you go through. At first, you will solve problems by writing out several steps. As you get better at doing that particular kind of math, you can go through other, similar examples of that kind more quickly and easily. As part of this, you tend to write out fewer intermediate steps in the process of solving it.

Some people find it very easy to do more and more of the steps in their head. Because of how their minds work, abstract math comes naturally to them. They also find relatively arcane math more accessible. That's not to say it isn't hard work for them - generally, it is! But some of the things they do inside serve to make the simple math easier, and allow the harder things to be possible.

These characteristics are based in common mental abilities that nearly every human has. In the following chapters, you will learn about some of these characteristics, and how to develop them in yourself.

Given an integral, to evaluate it you want to rearrange it into a form $\int f(u)\ du$, where f is some function whose antiderivative $F(u)$ is known to you. That's the abstract; concrete examples of this would be $f(u)=\sin u$ and $F(u)=-\cos u$, or $f(u)=1/u$ and $F(u)=\ln u$, or $f(u)=u^{2}+2u^{3}$ and $F(u)=\frac{1}{3}u^{3}+\frac{1}{2}u^{4}.$

When you learn to do integration in a calculus course, one of the first things you are taught is to manage complexity by doing variable substitution. Say you were integrating $\int\sin x^{2}\ 2x\ dx$; by inspecting the equation, you might think to try setting $u=x^{2}$, notice that $du=2x\ dx$, and rewrite the expression to be $\int\sin u\ du$. After evaluating it to $-\cos u+C$ (where C is a constant), you would plug back in to get $-\cos x^{2}+C$.

All the steps described above can be done easily enough on paper. We can do something quite similar mentally, in a way that scales to more complex integrands. A big key is to manage it in a way that you can visualize the expression as well as the processes of evaluating it.

If your visual imagination is very highly developed, you can do complex math mentally just as you would when solving it on paper. You could just imagine a sheet of paper (or whiteboard, chalkboard, etc.), and imagine all the symbols that you would write on the paper physically as you evaluated the expression. However, many people would have trouble doing this with larger expressions (or even with small ones). Even if you are able to visualize that well, there are better ways. A piece of paper is one medium; your imagination is a different one, and one that is more flexible. If you envision a whiteboard in your mind, with mathematical symbols written on it, you can do things with this mental representation that would be impractical or impossible if you were writing on a physical whiteboard. In fact, some of these things you can do are very powerful, and can more than make up for the fact that you don't have a written record in front of you.

As a human, there are a few mental resources you have that you can use. The first is your ability to visualize. Some people seem to be born with an ability to visualize well; others don't seem to consciously visualize at all. Most people are somewhere in between. If you do not feel very strong in this area, there is a visualization tutorial available at . (This tutorial is part of Inner Algebra, a book that teaches how to solve algebra equations mentally.)

Another mental resource you have is your memory. You can use your memory in a few specific ways that will help you manage the mental process you go through. A key one is called windowing. Rather than visualize the entire expression or equation, you visualize a portion of it that you want to work with. Later, you can visualize the rest of it or another part of it, recalling it from memory as you need to. This works because when working with an expression, you often only need to alter a few of the symbols; most of the expression will not be affected, at least for that one step. Windowing is based on the fact that most people can recall a math expression that is bigger than the largest one they can easily visualize. You can also do this with smaller expressions that you CAN easily visualize. In fact, it's highly recommended. If the expression is small enough that you can visualize the whole thing, you choose to visualize only part of it at any given moment. Then, when you need to work with another part of the expression, or the whole thing, you can consciously construct the image from your memory. Simple as it may sound, this can be tremendously helpful in practice, allowing you to do mental math much faster. (You can read a more in-depth discussion on windowing, also from Inner Algebra, at . That section also has some short but valuable exercises.)

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