Answer
$v=4t^2-\cot t -7-\pi^2$
Work Step by Step
We need to find the anti-derivative for $\dfrac{dv}{dt}= 8t+\csc^2 t$
Thus, we have $v=4t^2-\cot t +C$
Apply the initial condition $v(\frac{\pi}{2})=-7$ in the above equation to solve for $C$.
we get $\pi^2-0+C=-7 \implies C=-7-\pi^2$
Hence, $v=4t^2-\cot t -7-\pi^2$
