Problem $2\mathrm{.}2.$ In the following examples, find the position function $x(t)$ of a moving particle with given acceleration $a(t),$ initial position $x(0)=x_0$ and initial velocity $v(0)=v_0.$

a$)a(t)=-20,x_0=5,v_0=-15$

b$)a(t)=3t,x_0=0,v_0=5$

c$)a(t)=2t+1,x_0=4,v_0=-7$

d$)a(t)=4(t+3{)}^2,x_0=1,v_0=-1$

e$)a(t)=\frac{1}{\sqrt[\phantom{2}]{t+4}},x_0=1,v_0=-1$

f$)a(t)=\frac{1}{(t+1{)}^3},x_0=0,v_0=0$

g$)a(t)=50sin5t,x_0=8,v_0=-10$