Dark Secrets of Mathematicians: Something About Integration By Parts


A friend took me up last night on my offer to help with any mathematics she was unsure about. I’m happy to do it, though of course it came as I was trying to shut things down for bed. But that part was inevitable and besides the exam was today. I thought it worth sharing here, though.

There’s going to be some calculus in this. There’s no avoiding that. If you don’t know calculus, relax about what the symbols exactly mean. It’s a good trick. Pretend anything you don’t know is just a symbol for “some mathematics thingy I can find out about later, if I need to, and I don’t necessarily have to”.

“Integration by parts” is one of the standard tricks mathematicians learn in calculus. It comes in handy if you want to integrate a function that itself looks like the product of two other functions. You find the integral of a function by breaking it up into two parts, one of which you differentiate and one of which you integrate. This gives you a product of functions and then a new integral to do. A product of functions is easy to deal with. The new integral … well, if you’re lucky, it’s an easier integral than you started with.

As you learn integration by parts you learn to look ways to break up functions so the new integral is easier. There’s no hard and fast rule for this. But bet on “the part that has a polynomial in it” as the part that’s better differentiated. “The part that has sines and cosines in it” is probably the part that’s better integrated. An exponential, like 2x, is as easily differentiated as integrated. The exponential of a function, like say 2x2, is better differentiated. These usually turn out impossible to integrate anyway. At least impossible without using crazy exotic functions.

So your classic integration-by-parts problem gives you an expression like this:

\int x \sin(x) dx = -x \cos(x) - \int \sin(x) dx

If you weren’t a mathematics major that might not look better to you, what with it still having integrals and sines and stuff in it. But ask your mathematics friend. She’ll tell you. The thing on the right-hand side is way better. That last term, the integral of the sine of x? She can do that in her sleep. It barely counts as work, at least by the time you’ve got in class to doing integration by parts. It’ll be -x\cos(x) + \cos(x) .

But sometimes, especially if the function being integrated — the “integrand”, by the way, and good luck playing that in Scrabble — is a bunch of trig functions and exponentials, you get some sad situation like so:

\int \sin(x) \cos(x) dx = \sin^2(x) - \int \sin(x) \cos(x) dx

That is, the thing we wanted to integrate, on the left, turns up on the right too. The student sits down, feeling the futility of modern existence. We’re stuck with the original problem all over again and we’re short of tools to do something about it.

This is the point my friend was confused by, and is the bit of dark magic I want to talk about here. We’re not stumped! We can fall back on one of those mathematics tricks we are always allowed to do. And it’s a trick that’s so simple it seems like it can’t do anything.

It’s substitution. We are always allowed to substitute one thing for something else that’s equal to it. So in that above equation, what can we substitute, and for what? … Well, nothing in that particular bunch of symbols. We’re going to introduce a new one. It’s going to be the value of the integral we want to evaluate. Since it’s an integral, I’m going to call it ‘I’. You don’t have to call it that, but you’re going to anyway. It doesn’t need a more thoughtful name.

So I shall define:

I \equiv  \int \sin(x) \cos(x) dx

The triple-equals-sign there is an extravagance, I admit. But it’s a common one. Mathematicians use it to say “this is defined to be equal to that”. Granted, that’s what the = sign means. But the triple sign connotes how we emphasize the definition part. That is, ‘I’ might have been anything at all, and we choose this of the universe of possibilities.

How does this help anything? Well, it turns the integration-by-parts problem into this equation:

I = \sin^2(x) - I

And we want to know what ‘I’ equals. And now suddenly it’s easier to see that we don’t actually have to do any calculus from here on out. We can solve it the way we’d solve any problem in high school algebra, which is, move ‘I’ to the other side. Formally, we add the same thing to the left- and the right-hand sides. That’s ‘I’ …

2I = \sin^2(x)

… and then divide both sides by the same number, 2 …

I = \frac{1}{2}\sin^2(x)

And now remember that substitution is a free action. We can do it whenever we like, and we can undo it whenever we like. This is a good time to undo it. Putting the whole expression back in for ‘I’ we get …

\int \sin(x) \cos(x) dx = \frac{1}{2}\sin^2(x)

… which is the integral, evaluated.

(Someone would like to point out there should be a ‘plus C’ in there. This someone is right, for reasons that would take me too far afield to describe right now. We can overlook it for now anyway. I just want that someone to know I know what you’re thinking and you’re not catching me on this one.)

Sometimes, the integration by parts will need two or even three rounds before you get back the original integrand. This is because the instructor has chosen a particularly nasty problem for homework or the exam. It is not hopeless! But you will see strange constructs like 4/5 I equalling something. Carry on.

What makes this a bit of dark magic? I think it’s because of habits. We write down something simple on the left-hand side of an equation. We get an expression for what the right-hand side should be, and it’s usually complicated. And then we try making the right-hand side simpler and simpler. The left-hand side started simple so we never even touch it again. Indeed, working out something like this it’s common to write the left-hand side once, at the top of the page, and then never again. We just write an equals sign, underneath the previous line’s equals sign, and stuff on the right. We forget the left-hand side is there, and that we can do stuff with it and to it.

I think also we get into a habit of thinking an integral and integrand and all that is some quasi-magic mathematical construct. But it isn’t. It’s just a function. It may even be just a number. We don’t know what it is, but it will follow all the same rules of numbers, or functions. Moving it around may be more writing but it’s not different work to moving ‘4’ or ‘x2‘ around. That’s the value of replacing the integral with a symbol like ‘I’. It’s not that there’s something we can do with ‘I’ that we can’t do with ‘\int \sin(x)\cos(x) dx ‘, other than write it in under four pen strokes. It’s that in algebra we learned the habit of moving a letter around to where it’s convenient. Moving a whole integral expression around seems different.

But it isn’t. It’s the same work done, just on a different kind of mathematics. I suspect finding out that it could be a trick that simple throws people off.

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