a particle is moving along a straight line so that after t seconds after passing through a fixed point O on the line, its velocity is given by v(t)=t*sin((3^.5)*t)
a) find the values of t for which v(t)=0 given that 0 is less than or equal to t is less than or equal to 6

b) write down a mathematical expression for the total distance traveled by the particle in the particle in the first six seconds after passing through O. And find this distance.

a) You want v(t)=0, then



thus leading to t=0 or for some integer n: as you have imposed , n can assume here only values 0, 1, 2, 3 (n=4 corresponds to thus t>6).

Then you have

b) The distance (considering signs) traveled is



Integrating this expression you get



and substituting you get the answer.

If you want the distance without considering signs then you have to integrate




which is done in a similar manner, just remember that the primitive function of is