Question

the acceleration of particle which starts from rest varies with according to relation a=2t+3 the velocity of the particle at time t would be

Answer 1

Just integrate it.
Ans t^2+3t

Answer 2

a = 2t + 3

$a= \frac{d(v)}{dt}$
a dt = d(v)
integrate both the sides

In (2t + 3) dt = In dv
$\frac{2 {t}^{2} }{2} \: + 3t + c = v$
v = t² + 3t + c