A particle \(P\) of mass 0.4 kg moves along the \(x\)-axis in the positive direction. At time \(t = 0 , P\) passes through the origin \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds \(P\) is \(x\) metres from \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 8 } { ( t + 4 ) ^ { 2 } } \mathrm {~N}\) and is directed towards \(O\).
Show that \(v = \frac { 20 } { t + 4 } + 5\)
When \(v = 6 , x = a + b \ln 5\), where \(a\) and \(b\) are integers.
Using algebraic integration, find the value of \(a\) and the value of \(b\).