Compute the integral:

${\int }_0^1{x}^{11}{e}^{x^3}dx$

Find a formula for $\int x^k\arctan \left(x\right)dx$, where $k$ is a positive integer.

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}\sin x\cdot \ln \left(\cos x\right)dx$

The integration by parts formula is most closely related to which of the following derivative rules?

A study models the hourly rate of trucks passing a checkpoint as $R\left(t\right)=500t{e}^{-\frac{t}{4}}$ for $t$ in hours. How many trucks pass the checkpoint from $t=0$ to $t=8$? Round the answer to the nearest whole number.