Mathematics Asked by Jundan Luo on November 10, 2020

I am writing an academic paper. And I wonder which is the best English term for "the result of a mathematical integration".

For example, I have a mathematical integration as below.

$$F = int f(u) du$$

Which is a more suitable name for $F$? "integral $f(u)$" or "integrated $f(u)$"? I must name $F$ in a way expressing its relationship with $f(u)$.

Thanks for your comments! As it is the "integral of $f(u)$", can I name it "integral $f(u)$" for short? I think I have to omit the preposition when defining a new academic word. ($f(u)$ is an already-defined physical term.)

The notation $int f$ is generally used in one of two contexts: either it represents a *number* (which we generally interpret as the area under the graph of $f$, but it is actually more general than that), or it represents a *function*.

If $int f$ represents a number, then it is a definite integral. More commonly, this is written in the form
$$ int_{a}^{b} f(x) ,mathrm{d}x
qquadtext{or}qquad
int_E f(x) ,mathrm{d}x.$$
Sometimes, a definite integral will be written without giving explicit bounds for the integration, or without specifying a domain of integration. In such a context where a number is still meant, this notation is generally understood to indicate integration over the entire domain of the function. For example,
$$ int mathrm{e}^{-x^2},mathrm{d}x = int_{-infty}^{infty} mathrm{e}^{-x^2}, mathrm{d}x = sqrt{pi}.$$
In this context, the number which is obtained by integration is called the **integral**. More precisely, one might say that $int_E f(x),mathrm{d}x$ is called the "**definite integral** of $f$ over $E$".

The same notation is also used to denote a function. For example, if $f$ is a "sufficiently nice" read-valued function defined on the real numbers, and there is a function $F$ with the property that $F'(x) = f(x)$ for all $x$, then we say that $F$ is an **antiderivative** or **primitive** of $f$. This is often written as
$$ F(x) = int_{a}^{t} f(t),mathrm{d}t
qquadtext{or}qquad
F = int f. $$
It should be noted that $F$ is not unique—a function may have *many* antiderivatives, though these antiderivatives differ by only a constant, so it is not hard to specify the entire family of antiderivatives. The notation $int f$ is, perhaps, confusing, but it is justified by the Fundamental Theorem of Calculus, which demonstrates that definite integrals and antiderivatives are related, e.g. in the setting of Riemann integration, $F$ is an antiderivative of $f$, then
$$ int_{a}^{b} f(x),mathrm{d}x = F(b) - F(a). $$
This function might also be called the indefinite integral or inverse derivative. Wikipedia gives another couple of terms.

The notation $f$ denotes a function, while $f(u)$ or $f(x)$ denotes a *value* of that function—this latter notation represents a *number* (or dependent variable), not a function. Thus it is a little strange to talk about "the integral of $f(u)$". Thus it is fine to say that something is the "integral of $f$", but it is not quite right to say that it is the "integral of $f(u)$".

If $int f$ denotes a number, that number is the **integral** or **definite integral** of $f$ (over some interval or domain). If $int f$ denotes a function, that function is an **antiderivative** or **primitive** of $f$. In either case, I would not elide the preposition—keep that "of" in there.

Answered by Xander Henderson on November 10, 2020

I agree with some comments. The term "integral" is used both for the problem and for the answer. This is like a lot of other words in mathematics:

5+3 is an easy sum

What is 5+3? The sum is 8.

5! is the product of the numbers from 1 to 5

120 is the product of the numbers from 1 to 5

Answered by GEdgar on November 10, 2020

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