When you integrate a function - for instance, $\cos(3x)$, you probably have to stop for a moment and think: “Do you multiply by 3 or divide when you integrate?” Some people don’t even get that far, and just say “Oh, it must be $\sin(3x)$”, and all of us can just sigh at them.

The answer is that you divide when you integrate, and here are three reasons why.

(Note: this applies only when you have something linear in the bracket: that is, something like $ax + b$, where $a$ and $b$ are constants, and $a \ne 0$. If it’s a more complicated function, you need more complicated ideas).

1: You divide when you integrate because you multiply when you differentiate

This is a bit of a cheaty answer, but it’s one that helps me remember: when you differentiate $\sin(3x)$, you get $3 \cos(3x)$, so it follows that you’d differentiate $\frac{1}{3} \sin(3x)$ to get $\cos(3x)$. Because differentiation is the reverse process of integration, that means $\int \cos(3x) \dx = \frac{1}{3} \sin(3x) + c$.

2: You divide when you integrate because substitution says so

Oh yeah, hold it with your groaning. Substitution is the finest method of integration known to man, and one day I’ll teach you how to do it. Today I’m just going to blast through it.

So, you want to work out $\int \cos(3x) \dx$? Well, I don’t like the look of the $3x$. I’m going to call that $u = 3x$. However, you can’t integrate $u \dx$, because $u$ and $x$ are different variables. You need to turn the $\dx$ into something related to $\du$.

That’s easy, though: you can differentiate to say $\frac{\du}{\dx} = 3$, so $\dx = \frac{\du}{3}$((This meddling with $\dx$ as if it’s a variable… I don’t think it’s 100% kosher. However, it works fine in practice, just not in theory.)). That means you now have $\int \cos (u) \frac{\du}{3}$. But that’s easy, too: you get $\frac{1}{3} \sin(u) + c$, only we made $u$ up; let’s turn it back into a $3x$ out of good manners, and get $\frac{1}{3} \sin(3x) + c$. Perfect.

3: The real reason you divide when you integrate (it’s to do with areas)

Think back to bad guy $x$. You remember, of course, that bad guy $x$ is in bracket prison because he does the opposite of what he’s told. If you tell him to be three times as big, he shrinks horizontally and become a third of the size.

So, having $(3x)$ in the bracket means that the graph would be a third as wide - meaning the area shrinks by the same factor. That means, the integral of $\cos(3x)$ - even without working it out - is a third of the size of the integral of $\cos(x)$. The same goes for any function!

Have you got any better explanations of why you divide by the number in front of the $x$?

* Edited 2021-06-16 to improve LaTeX