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(B) The work done $W$ by a variable force $F$ acting on a particle as it moves from position $x_i$ to $x_f$ is given by the integral: $W = \int_{x_i}^

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$\frac{1}{2} C{x_1}^2$

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(B) The work done $W$ by a variable force $F$ acting on a particle as it moves from position $x_i$ to $x_f$ is given by the integral: $W = \int_{x_i}^{x_f} F \, dx$. Given the force $F = Cx$ and the limits $x_i = 0$ to $x_f = x_1$,we substitute these into the integral: $W = \int_{0}^{x_1} Cx \, dx$. Since $C$ is a constant,we can take it out of the integral: $W = C \int_{0}^{x_1} x \, dx$. The integral of $x$ with respect to $x$ is $\frac{x^2}{2}$. Applying the limits: $W = C \left[ \frac{x^2}{2} \right]_{0}^{x_1} = C \left( \frac{{x_1}^2}{2} - \frac{0^2}{2} \right) = \frac{1}{2} C{x_1}^2$. Thus,the work done is $\frac{1}{2} C{x_1}^2$.

$A$ particle moves from $x = 0$ to $x = x_1$ under the influence of a force $F = Cx$. Find the work done in this process.

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