A particle moves along the positive \(x\)-axis with a constant speed \(v_0\). Starting at time \(t = 0\) at position \(x = 0\), the particle experiences a force parallel to its velocity given by \(F(x,t) = k_0 e^{-\alpha t} x\), where \(k_0\) and \(\alpha\) are positive constants. Which of the following expressions represents the total work done on the particle by this force from \(t = 0\) to \(t \to \infty\)?