Is it required to use brackets inside an integral?

Solution 1:

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.)

Solution 2:

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.