Answer
$\left\langle\ln 4, \frac{56}{3 }, -\frac{496}{5 } \right\rangle.$
Work Step by Step
We have
\begin{align}
\int_{1}^{4}\left\langle t^{-1},4 t^{1/2}, -8 t^{3/2} \right\rangle dt &=\left\langle \ln t, \frac{4}{3/2} t^{3/2},-\frac{8}{5/2} t^{5/2}
\right\rangle|_{1}^{4}\\
&=\left\langle \ln 4, \frac{8}{3} 4^{3/2},-\frac{16}{5} 4^{5/2}\right\rangle-\left\langle 0, \frac{8}{3} ,-\frac{16}{5} \right\rangle\\
&=\left\langle\ln 4, \frac{56}{3 }, -\frac{496}{5 } \right\rangle.
\end{align}
