Notation for antiderivative I came across the following:
$$
F(x) = \int x^3 \cos(x)dx
$$
where $F$ is understood to be a primitive of $x^3 \cos(x)$. I find this confusing, because of the "same" $x$ appearing on both sides of the equality. To me, $x$ is "integrated out" on the right side, and I prefer the notation:
$$ 
F(x) = \int_{0}^{x} u^3 \cos(u) du
$$
or possibly:
$$ 
F = \int x^3 \cos(x) dx
$$
without mentioning the variable for F.
Is the first notation widely used?

The answer depends on what $\int f(x) \,dx $ means, about which there is no universal agreement. One interpretation is: the set of all functions whose derivative is $f$. If this definition is accepted, then $=$ should really be read as $\in$. This is the same convenient abuse of notation as in $\sqrt{x^2+1}=O(x)$. The other abuse is in writing $F(x)$ when you mean $F$, and this is also convenient at times. 
So, this is how $F\in \int x^3\,dx$  becomes $F(x)=\int x^3\,dx$.
Notice that there is no integration involved in the above interpretation. 
2nd interpretation: Someone may say that $\int f(x)\,dx$ is actually an integral, namely $\int_a^x f(t)\,dt$ with unspecified $a$. If you subscribe to this point of view, then $\sin x=\int \cos x\,dx$ is a true statement while $\sin x+5=\int \cos x\,dx$ is false. 
