The instantaneous velocity of a particle moving in a straight line is given by   $V=\alpha t+\beta t^2$  where ‘ $\alpha$’ and ‘$\beta$ ’ are constants the distance travelled by the particle between 1 sec and 2 sec is: 

Given  that  $V=\alpha t+\beta t^2$

Integrating on both sides

 $\underset{S1}{\overset{S2}{\int }}ds=\underset{1}{\overset{2}{\int }}(\alpha t+\beta t^2)dt$

${\left[S\right]}_{s1}^{s2}={{\left[\frac{\alpha t^2}{2}+\beta \frac{t^3}{3}\right]}_1}^2$   

  $[S_2-S_1]=\frac{\alpha }{2}[2^2-1^2]+\frac{\beta }{3}[2^3-1^3]$  
 
 $=\frac{\alpha }{2}(4-1)+\frac{\beta }{3}[8-1]$  

  $S_2-S_1=\frac{3\alpha }{2}+\frac{7\beta }{3}$

As the particle move along a straight line  

Distance = displacement 
Distance $=\frac{3\alpha }{2}+\frac{7\beta }{3}$