Answer
$4\sqrt{3}-3$
Work Step by Step
\begin{aligned} \int_{-\pi / 3}^{-\pi / 4}\left(4 \sec ^{2} t+\frac{\pi}{t^{2}}\right) d t&=\int_{-\pi / 3}^{-\pi / 4}\left(4 \sec ^{2} t+\pi t ^{-2}\right) d t\\
&=\left(4 \tan t-\pi t ^{-1}\right)\bigg|_{-\pi / 3}^{-\pi / 4}\\
&= 4(\tan(-\pi/4)-\tan(-\pi/3))-\pi(\frac{-4}{\pi}+\frac{3}{\pi})\\
&=4\left(-1+\sqrt{3}\right)+1\\
&=4\sqrt{3}-3 \end{aligned}