Which is the best English term for "the result of a mathematical integration"? I am writing an academic paper. And I wonder which is the best English term for &quot;the result of a mathematical integration&quot;.
For example, I have a mathematical integration as below.
$$F = \int f(u) du$$
Which is a more suitable name for $F$? &quot;integral $f(u)$&quot; or &quot;integrated $f(u)$&quot;? I must name $F$ in a way expressing its relationship with $f(u)$.

Thanks for your comments! As it is the &quot;integral of $f(u)$&quot;, can I name it &quot;integral $f(u)$&quot; for short? I think I have to omit the preposition when defining a new academic word. ($f(u)$ is an already-defined physical term.)
I agree with some comments.  The term &quot;integral&quot; 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
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 &quot;definite integral of $f$ over $E$&quot;.
A function
The same notation is also used to denote a function.  For example, if $f$ is a &quot;sufficiently nice&quot; 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 &quot;the integral of $f(u)$&quot;.  Thus it is fine to say that something is the &quot;integral of $f$&quot;, but it is not quite right to say that it is the &quot;integral of $f(u)$&quot;.
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 &quot;of&quot; in there.