The work done by a variable one-dimensional force $F(x)$ in displacing a particle from an initial position $x_i$ to a final position $x_f$ is given by the definite integral of the force with respect to displacement: $W = \int_{x_i}^{x_f} F(x) \, dx$

Given that the force is $F = Cx$, the initial position is $x_i = 0$, and the final position is $x_f = x_1$. Substituting these values into the formula, we get: $W = \int_{0}^{x_1} Cx \, dx$ $W = C \left[ \frac{x^2}{2} \right]_{0}^{x_1}$ $W = C \left( \frac{x_1^2}{2} - \frac{0^2}{2} \right) = \frac{1}{2} C x_1^2$

Thus, the work done in the process is $\frac{1}{2} C x_1^2$.