TITLE: Is it required to use brackets inside an integral?
QUESTION [27 upvotes]: Is it necessary for someone to write
$$\int (x^2+2x) \,\mathrm{d}x$$
Instead of 
$$\int x^2+2x \,\mathrm{d}x$$
With the second one, it's quite obvious which terms we are taking the integral of. Is it still necessary to use brackets in this case?

REPLY [3 votes]: The short answer is, yes, you must use the parentheses.
The $dx$ of the integral is not just a delimiter to mark the end of the integral. $dx$ acts somewhat like a variable itself, which the integrand is "multiplied" with.
For example, this:
$$\int \frac{1}{x} \, dx$$
Can equivalently be written as:
$$\int \frac{dx}{x}$$
And you will occasionally see such usage. So long as the expression can be rewritten using the rules of algebra so that the $dx$ is on the right and it is "multiplied" with everything before it, it is a valid integral.
If you write:
$$\int x^2 + 2x \, dx$$
Then you are only "multiplying" the $dx$ with $2x$, not $x^2+2x$, so you do not have a valid integral. People will still know what you mean, but in the same sense people know what you mean if you write "I can haz cheezburger?" — it doesn't make it proper English.
Why treat $dx$ as a variable and do funny things with it? It makes sense when doing differential equations. For example, if you treat $dx$ and $dy$ and variables, then this:
$$\frac{dy}{dx} = xy$$
Can be rewritten using algebra as:
$$\frac{dy}{y} = x \, dx$$
After which you can integrate both sides:
$$\int \frac{dy}{y} = \int x \, dx$$
And you will get the algebra equation $\ln |y| = \frac{1}{2}x^2 + C$. It looks crazy, but this is a standard technique for solving differential equations.