Which is the best English term for "the result of a mathematical integration"?

Question

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 $f(u)$.

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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.)

Answer 1

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

Answer 2

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.

A number

If $\int f$ represents a number, then it is a definite integral. More commonly, this is written in the form $$ \int_{a}^{b} f(x) \,\mathrm{d}x \qquad\text{or}\qquad \int_E f(x) \,\mathrm{d}x.$$ 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 $E$".

A function

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 $$ F(x) = \int_{a}^{t} f(t)\,\mathrm{d}t \qquad\text{or}\qquad F = \int f. $$ 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.

A note on notation

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 $f(u)$".

TL;DR

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.