Answer
$s(t) = -3cost + 2sint + 2t + 3$
Work Step by Step
Given:
$a(t)=3cos t-2sin t$
Thus we have:
$v(t) = 3sint - 2(-cost)+C$
$v(t) = 3sint + 2cost + C$
$v(0) = 3sin(0) + 2cos(0) + C$
$v(0) = 3(0) + 2(1) + C$
Given: $v(0) = 4$
$4 = 2+ C$
$2 = C$
Plug in the value of $C$ in the function of $v(t)$.
$v(t) = 3sint + 2cost + 2$
$s(t) = 3(-cost) + 2sint + 2t + C$
$s(t) = -3cost + 2sint + 2t + C$
$s(0) = -3cos(0) + 2sin(0) + 2(0) + C$
$s(0) = -3(1) + 2(0) + 2(0) + C$
$s(0) = -3 + C$
Given $s(0) = 0$
$0 = -3 + C$
$C = 3$
Plug in the value of $C$:
$s(t) = -3cost + 2sint + 2t + 3$
