A particle moves along the \(x\)-axis such that its acceleration for \(t \ge 0\) is given by \(a(t) = \beta t^2 - c\), where \(\beta\) and \(c\) are positive constants. At time \(t = 0\), the particle is located at position \(x(0) = x_0\) and has velocity \(v(0) = v_0\). Which of the following integral expressions correctly gives the position \(x(t)\) of the particle as a function of time \(t\)?