# Question #25ae1

Jul 14, 2016

It helps clarify what you're integrating, exactly.

#### Explanation:

The $\mathrm{dx}$ is there, for one, by convention. Recall that the definition of definite integrals comes from a summation that contains a $\Delta x$; when $\Delta x \to 0$, we call it $\mathrm{dx}$. By changing symbols as such, mathematicians imply a whole new concept - and integration is indeed very different from summation.

But I think the real reason why we use $\mathrm{dx}$ is to clarify that you are indeed integrating with respect to $x$. For example, if we had to integrate ${x}^{a}$, $a \ne - 1$, we would write $\int {x}^{a} \mathrm{dx}$, to make it clear that we are integrating with respect to $x$ and not to $a$. I also see some sort of historical precedent, and perhaps someone more versed in mathematical history could expound further.

Another possible reason simply follows from Leibniz notation. We write $\frac{\mathrm{dy}}{\mathrm{dx}}$, so if $\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{x}$, for example, then $\mathrm{dy} = {e}^{x} \mathrm{dx}$ and $y = \int {e}^{x} \mathrm{dx}$. The $\mathrm{dy}$ and $\mathrm{dx}$ help us keep track of our steps.

However, at the same time I do see your point. To someone with more experience than average in calculus, $\int 3 {x}^{2}$ would make as much sense as $\int 3 {x}^{2} \mathrm{dx}$; the $\mathrm{dx}$ in those situations is a bit redundant. But you can't expect only those people to look at the problem; students starting out in the subject are more comfortable with a little more organization in the problem (at least from my experience), and I think the $\mathrm{dx}$ provides that.

I am positive there are other reasons why we might use $\mathrm{dx}$ so I invite others to contribute their ideas.