Question

Given acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t.

a = 10cos5t ; v(0) = -6 ; s(0) = 12

Answer 1

integrate a(t) from 0 to t
the constant of integration will be v(0)
integrate again from 0 to t
the constant of integration will be s(0)

Answer 2

It seems take you forever to understand the purpose of initial values :/
V(t) = ∫ a(t) dt
= 2sin 5t + C
V(0) = 2 sin 0 + C = - 6
-> C = -6
=>S(t) = ∫ v(t)dt = ∫ (2sin5t -6 ) dt
= .... + C
S(0) = 12 -> C = ...