Is it required to use brackets inside an integral? 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?
 A: 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.
A: I disagree with the other answers here.
$$\int x^2+2x \,\mathrm{d}x$$
is not correct; the integral needs to be written as
$$\int (x^2+2x) \,\mathrm{d}x$$
instead.
Think of the definite integral, which is really the source of this notation — the definite integral here would be a limit of sums of the form
$$\sum_k (x_k^{\,2}+2x_k) \,\Delta x,$$
not sums of the form
$$\sum_k x_k^{\,2}+2x_k\Delta x.$$
The standard notation works for integrals because you can treat the integral as similar to a summation, and you can treat the part after the integral sign as similar to a product of the integrand and $\mathrm{d}x.$  (Obviously this is just a similarity, not a rigorous definition, but it works in practice.)
Here's an example where it matters: If you want to use a change of variables and apply the substitution rule, you'll get the right answer if you start with 
$$\int (x^2+2x) \,\mathrm{d}x$$
and apply the usual laws of algebra, but you will not get the right answer if you start with
$$\int x^2+2x \,\mathrm{d}x$$
instead. (You'll need to add the parentheses back in which should have been there all along.)
For those people who think otherwise, look in published math textbooks or journals and see what kind of usage you find. (If actual usage is different, I would certainly acknowledge that, along with a suggestion then that people should use parentheses when needed to treat this formally as a product of the integrand and $\mathrm{d}x,$ for the reasons I've stated.)
A: By definition of how the notation works when you use the $\displaystyle\int f(x)\,dx $ method, anything between the integral sign and the $dx$ is considered the integrand. If you subscribe to the less-popular school of using the $\displaystyle\int \,dx(f(x))$ then you will need parentheses.
A: As others have stated it is not required but it is appealing to the eye as:
$$\int 2x+3\int x^2+2xy \,\mathrm{dx}\,\mathrm{dy}$$
is hard to read and understand...at least for me. Also as Bye_World stated below, it is hard to determine which variable to integrate in which part.
You can see how messy this double integral is and I might also add that triple integrals are quite common in most of the subjects like 3D mechanics and Differential geometry and many many things.
