Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function
\[
D^{\prime}(t)=850.89+828.42 t-159.33 t^{2}+11.64 t^{3}
\]
where $t$ is the number of years since 1995. By how much did the debt increase between 1996 and 2000 ?

Step 1 ：The integral of the function \(D'(t)\) from 1 to 5 is: \[\int_{1}^{5} D^\prime(t) dt = \int_{1}^{5} (850.89+828.42t-159.33t^{2}+11.64t^{3}) dt\]

Step 2 ：We can integrate term by term to get: \[\int_{1}^{5} 850.89 dt + \int_{1}^{5} 828.42t dt - \int_{1}^{5} 159.33t^{2} dt + \int_{1}^{5} 11.64t^{3} dt\]

Step 3 ：The integral of each term is: \[850.89t \Big|_{1}^{5} + 414.21t^{2} \Big|_{1}^{5} - 53.11t^{3} \Big|_{1}^{5} + 2.91t^{4} \Big|_{1}^{5}\]

Step 4 ：Evaluating these at the limits of integration gives: \[(850.89*5 - 850.89*1) + (414.21*5^{2} - 414.21*1) - (53.11*5^{3} - 53.11*1) + (2.91*5^{4} - 2.91*1)\]

Step 5 ：Solving this gives: \[4254.45 + 10355.25 - 6631.25 + 1812.5 - 850.89 - 414.21 + 53.11 - 2.91\]

Step 6 ：Adding these up gives: \[8106.24\]

Step 7 ：So, the national debt increased by \(\boxed{8106.24}\) billion (or \(\boxed{8.10624}\) trillion) between 1996 and 2000.