We have a Curve $C:\vec{x}\left(t\right)=\begin{pmatrix}1-2t^2\\ t\end{pmatrix}$ Now you have to calculate $\int _C\vec{F}\left(\vec{x}\right)d\vec{x}$ for
$\vec{F}\left(\vec{x}\right)=\begin{pmatrix}1\\ 0\end{pmatrix}$
$\vec{F}\left(\vec{x}\right)=\begin{pmatrix}x_2\\ x_1\end{pmatrix}$
There are many more practice problems just like this and I'd like to try and solve them but I don't know how to start.
Do I make $1-2t^2 =1$ or $t=0$?
What does $\vec{F}\left(\vec{x}\right)=\begin{pmatrix}x_2\\ x_1\end{pmatrix}$ even mean?
Note: if you dont know the notation $f\circ g$ just mean the composition of functions $f$ and $g$, that is, $(f\circ g)(s)=f(g(s))$.
There is something that could confuse you: the symbol $x$ is used for two different purposes in the exercise
to denote a vector on the domain of $F$, and
to denote a function from $\Bbb R\to\Bbb R^2$.
Just substituting we have that
$$(F\circ x)(t)=F(1-2t^2,1)$$
where we assumed here that the domain of $F$ is some subset of $\Bbb R^2$ (probably $\Bbb R^2$ itself) because the image of the function $x$ is in $\Bbb R^2$.
Now, for the first case we have that $(F\circ x)(t)=(1,0)$, that is, every $t\in\Bbb R$ defines some $x(t)\in\Bbb R^2$, thus $F(x(t))=(1,0)$ because $F$ is already constant.
For the second case we have that $(F\circ x)(t)=(t,1-2t^2)$, assuming that the notation $x_k$ refer to the coordinates of the vector $x$.
Can you evaluate $\int_C F(x)\cdot dx$ in each case now?