Question

Find the velocity, v(t), for an object moving along the x-axis if the acceleration, a(t), is a(t) = 3t^2 + cos(t) and v(0) = 2.

Answer 1

$\bf \displaystyle a(t)=3t^2+cos(t)\implies \stackrel{velocity}{\int~ [3t^2+cos(t)]dt}\\\\\\ t^3+sin(t)+C=~~~v(t) \\\\\\ \begin{cases} v(0)=2\\ v=2\\ t=0 \end{cases}\implies 0^3+sin(0)+C=2\implies C=2 \\\\\\ \boxed{t^3+sin(t)+2=v(t)}$