Shifting Velocity and Position functions

Question

I'm given a function $A(t)$ that defines the acceleration of an object w.r.t. time $t$ and am tasked with finding the position function and velocity function for that object.

Finding the functions seems simple enough; take the Derivative to find $P(t)$ and Integrate to find $V(t)$. However I'm given that the initial position of the object is $0$ and the initial Velocity is $5$, and the value of the Derivative/Integral of $A(t)$ at $t=0$ is $-4$ and $4$ respectively (as opposed to $0$ and $5$).

The functions:

$A(t)=2.6+(t-4)sin(t)$

$P(t) = A'(t) = (4 - t)cos(t) + sin(t)$

$V(t) = \int A(t)dt = 2.6t+4cos(t) - tcost(t) - sin(t)$

My question is: Is it sufficient to apply a basic vertical/horizontal shift to the Position and Velocity functions to get an accurate model, or will I need to take other steps to get the correct functions?

In other words, are the following adjustments valid or erroneous?

$P(t) = ((4 - t)cos(t) + sin(t)) + 4$

$V(t) = 1 + 2.6t + 4cos(t) - tcost(t) - sin(t)$

Answer 1

The formulas you need to apply are: $V(t)=V(0)+∫t0A(t)dt$ and then $P(t)=P(0)+∫t0V(t)dt$ (assuming that V is velocity and P is position). The initial conditions give you $P(0)=0$ and $V(0)=5$. –

-Winther