Answer
$-\dfrac{8}{3}\cot^3t+8\cot t+8t+C$
Work Step by Step
Use the Reduction Formula:
$\int\cot^naxdx=-\dfrac{\cot^{n-1}ax}{a(n-1)}-\int\cot^{n-2} (ax) \space dx$
Let $I=8[-\dfrac{\cot^3t}{3}-\int\cot^2tdt ] \\=8[-\dfrac{\cot^3t}{3}-(-\dfrac{\cot t}{1}-\int\cot^0tdt\Big)]\\=8[-\dfrac{\cot^3t}{3}+\cot t+t]+C\\=-\dfrac{8}{3}\cot^3t+8\cot t+8t+C$
