Find the displacement function if the acceleration function is a=2t+5. Assume a zero initial condition of displacement and v=8 when t=1.

Integrating the acceleration function gives the velocity function v, as below:
v = t² +5t +C₁, where C₁ is a constant.

Integrating the velocity function gives the displacement function x, as below:
x = t³/3 + 5t²/2 + C₁t + C₂, where C₂ is another constant.

The answer is completed by finding the 2 constants, C₁ and C₂.

With a zero initial condition of displacement, that means t=0, x=0. Put this initial condition into the displacement function ---> C₂ = 0.

The boundary condition is that: v=8 when t=1. Simply put this condition into the velocity function ---> C₁ = 2.

Thus, the complete displacement function is as below:
x =  t³/3 + 5t²/2 + 2t