A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t \mathrm{~s}\) the velocity of \(P\) is \(v \mathrm{~m} \mathrm{~s}^{-1}\) and the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm{~m}\). The only force acting on \(P\) is a resistive force of magnitude \(\left(4 \mathrm{e}^{-x}+12\right) \mathrm{e}^{-x} \mathrm{~N}\). When \(t=0, x=0\) and \(v=4\).
(a) Show by integration that \(v=\frac{1+3 \mathrm{e}^{x}}{\mathrm{e}^{x}}\).

(b) Find an expression for \(x\) in terms of \(t\).