Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 62

You need to remember the relation between acceleration, velocity and position, such that:
int a(t)dt = v(t) + c
int v(t) dx = s(t) + c
You need to find first the velocity function, such that:
int (3cos t - 2sint )dt = int 3cos t dt - int 2sin t dt
int (3cos t - 2sint )dt = 3sin t + 2cos t + c
The problem provides the information that v(0) = 4, hence, you may find c:
v(0) = 3sin 0 + 2cos 0 + c => 4 = 0 + 2 + c => c = 2
Hence, the velocity function is v(t) = 3sin t + 2cos t + 2
You need to evaluate the position function, such that:
int (3sin t + 2cos t + 2) dt = int 3sin t dt + int 2cos t dt + int 2dt
int (3sin t + 2cos t + 2) dt = -3cos t + 2sin t + 2t + c
Hence, the position function s(t) = -3cos t + 2sin t + 2t + c can be completely determined, if the information provided by the problem is used, s(0) = 0.
s(0) = -3cos 0 + 2sin 0 + 0 + c => 0 = -3 + c => c = 3.
Hence, evaluating the position of the particle, yields s(t) = -3cos t + 2sin t + 2t + 3.