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How do I find displacement of x, when it is a function of t? How do I start this question?

The motion of a particle on a straight line has its velocity $v$ given as a function of time $t$ by $v(t)= 6cos(2t)$. Assume that at time $t=0$, the particle is at position $x=1$. Find the expression for the displacement $x$ as a function of time $t$ and show that the motion is restricted to the interval $-2\leq \!\, x \leq \!\,4$.

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1 Answer

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As $$v(t)=\frac{dx}{dt}=6\cos(2t),$$

we get

$$x(t)=x(0)+\int_0^t 6\cos(2\theta)d\theta $$

$$=1+3\sin(2t).$$

but

$$\forall t\in\mathbb R\; -3\leq 3\sin(2t)\leq 3,$$

then

$$\forall t \;\;1-3\leq x(t) \leq 1+3$$

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