A particle moves along the x-axis according to $x=A[1+\sin \omega t]$. What distance does is travel in time interval from t=0 to $t=2.5\pi /\omega$?

$$\begin{gathered} t=\frac{25\pi }{\omega }=\frac{2.5\pi }{(2\pi /T)} \\ =(2.5)\left(\frac{T}{2}\right)=5\left(\frac{T}{4}\right) \end{gathered}$$

Particle starts from extreme position and in every $\frac{T}{4}$ time it travels a distance A. So, in time t=5(T / 4) it will travel a distance 5 A.