Doing Calculus In Your Head: Case Study - Integration By Parts

Let's look at how you can learn to do calculus mentally, by carefully examining a specific example: evaluating . Integrating this requires doing integration by parts.

(Prerequisite: To understand what follows, you will need to know how to solve this on paper. In particular, you will need to be comfortable applying the integration by parts rule, .)

To start, you want to be able to create a mental image you will work with. Visualize this in your mind's eye:

If you can close your eyes and see the whole integral expression, you are ready. If not, practice it until you get it. A visualization tutorial is available in the Hilomath store. (A good quick way to practice: get a blank sheet of paper, and repeatedly try to write out the expression from memory. Just before you write it out each time, close your eyes and attempt to visualize it completely.)

In nearly every modern calculus textbook, integration by parts is taught using the symbols "u" and "v". So much that you probably think in terms of them. Let's take advantage of that. Visualize the equation again. We want to divide the integrand into parts u and v, and we want to mentally keep track of it. We'll do this by visualizing labeled boxes around them:

This is what you want to construct in your mind's eye. You can visualize it this way, and there are other ways to visualize it as well. For example, instead of seeing a floating "u" and "dv" over boxes, you may see the symbols corresponding to "u" be red, and those corresponding to "dv" be blue, like this:

We'll do it the first way in this essay. Experiment, to find the method that works best for you.

Now we're ready to apply the integration by parts rule.

Visualize this. Now switch back to the problem statement - visualize this:

It is important to be able to "switch" your attention back and forth rapidly - to change what you see in your mind's eye. Practice visualizing one, clearing your mind, then visualizing the other. Kind of like if you were swapping out small cards carrying the expressions in front of you.

We need three parts - u, v, and du - and we already have one (u = x). To manage things, start by visualizing the integration by parts rule.

One at a time, you are going to replace each symbol. We have u already, so let's substitute that:

Again, this is what you see in your mind's eye. Practice this now until you can repeatedly do it.

(Notice also how you are not visualizing the left-hand side of the integration-by-parts rule. That is important; focusing on only the relevant information helps us think more quickly and clearly. We don't need to visualize everything all the time.)

In this image, you have three symbols left to fill out: the v, twice, and the du. Recall that in the original integral, dv is equal to :

Mentally integrate dv to get a value for v, then plug it in. Do this by first "extracting" dv (), and visualizing it alone:

Then integrate the differential:

In your mind's eye, call up the expression again, plugging in for v:

Then get du by visualizing the expresion (which is simply "x", in this case), differenting it, and plugging in:

To simplify, we need to evaluate the new integral. Start by visualizing it by itself:

In your mind's eye, this is what you see. The whole expression is , but you don't visualize the part. It's not relevant at this moment; so you will conserve your mental energy by temporarily forgetting it. This makes it easier to evaluate that integral in your imagination:

becomes

This done, visualize the whole expression again:

If you made it this far, you will find it easy to simplify the rest of the way mentally:

Ignoring the constant of integration, this is the solution of the integral .

Representing Math in your Mind

After you have taken an introductory course in algebra or calculus, you can solve simple equations like "x+2=3" just by looking at them. If a problem is more complex, you may need to write out some steps on paper to finish it. Whether you are able to solve a given problem mentally depends a lot on the internal representation you use. Want to learn something interesting about yourself? Solve "x+2=3" in your head right now, observing how you do it. Do you see the equation in your mind's eye? Do you see the number two moving to the other side, where it is subtracted from the three? That image of the equation is your internal representation. Most people good at math seem to actually visualize in a way I just described. They see the symbols of the equation in their mind's eye, and work out the problem by transforming that image. You can expand your math ability by representing the problem in a different way. Consider the equation (2-x)/(x+3)=2. Let's say you would have trouble solving this in your head. If that is not the case with you, just substitute a more complicated example. To solve the equation mentally, you need to have a good representation (the visual image), plus the ability to manipulate the image (by moving the symbols around, in your imagination), in a way that maintains mathematical correctness (of course!). Correctness is critical. And it's related to how you represent the image, because if you do not correctly remember where each symbol is, all is lost. It's easy with "x+2=3", which has just a few symbols. It's not so easy with an equation that has several dozen. The equation "(2-x)/(x+3)=2" has about a dozen symbols. But you don't have to represent it internally the same as if you were writing it out. Try this: visualize it, but let the "(2-x)" become a fuzzy blob, as if it is going out of focus. At the same time, let "(x+3)" become a different fuzzy blob. Can you see this? You have the ratio of two blobs equated to "2". You'll find this easier to visualize, because you are effectively dealing with fewer symbols. Of course, you can only go so far with fuzzy blobs! That's fine; you can solve the problem partway just with the blobs. When it's time, you focus back in on a blob, to work with its individual symbols. The point of turning a collection of symbols into blob is to make it easier to work with them as a single unit, when that is appropriate. In Inner Algebra, I call the process of making these blobs and working with them chunking. Try playing with different representations when you think mathematically. As you discover new ways to encode the problem, your skill with abstract math will grow.

Visualization and Math Ability

As you study people who are good at math, you find that they have cultivated certain foundational skills. One of the most important is visualization. Think of the last time you saw something hilarious. Got it? Okay, now examine how it was that you brought up that memory. Did you see, in your mind's eye, what it was that made you laugh? That's visualization - when you create a picture internally. Or, perhaps you did not see anything. That's fine - let's explore a little deeper. We all have a capacity to represent our thoughts in terms of different senses. And we all tend to do it in our own unique way. Some of us will tend use the visual sense more. When we remember something, we'll usually see an image inside. Some of use will instead represent it as a sound - rather than seeing something, we will hear it. Or feel, smell, taste, etc. In reality, most of us will tend to use several of these sensory channels to represent a thought or memory. And with practice, you can learn to expand your ability to make use of each. Try this: Imagine yourself smiling real big, genuinely happy - like you are looking at yourself in a mirror. Now. You may have found this easy to do. Or, you may have found it hard, or could not even do it. Regardless, you just got feedback about your current visualization ability. And you can benefit by increasing that ability. Visualization is so important, that in a math book I wrote, Inner Algebra, I spend a whole chapter discussing it. How can you increase your ability to visualize? (Hint: 90% of it is practice and just trying.) As you experience this increase in ability, do you find that abstract math gets easier?