2026 MAY TZA P1 Q3

A particle \(P\) is travelling along a straight line, with displacement measured relative to a point \(\mathrm{O}\) on the line. The velocity \(\mathrm{v}\) m s\(^{-1}\), of the particle at the time \(t\) seconds is given by

\(\mathrm{v} = \frac{1}{t + 1} + \mathrm{e}^{-t}\), where \(t \geq 0\).

1. Find \(\int \left( \frac{1}{t + 1} + e^{-t} \right) \mathrm{d}t\).

   [2]

Initially the particle is at \(\mathrm{O}\).

2. Find an expression for the displacement, \(s\) metres, in terms of \(t\).

   [3]