Find

$$\int 2x^2{e}^{-4x}dx$$

writing the answer in simplest form.

The rate of change of the volume of water in a reservoir, $V$ (in megalitres per hour), is modeled by the equation

$$\frac{dV}{dt}=\frac{10t+2}{(t-1)(2t+1)},t>1$$

where $t$ is the time in hours since a valve was opened. Use partial fractions and algebraic integration to show that the total change in volume between $t=2$ and $t=4$ is $\ln k$ megalitres, where $k$ is a fully simplified rational constant to be found.