Answer
$v=4t^2-\cot t -7-\pi^2$
Work Step by Step
The anti-derivative for $\dfrac{dv}{dt}= 8t+\csc^2 t$ is $v=4t^2-\cot t +C$
Now, apply the initial conditions $v(\frac{\pi}{2})=-7$.
This implies $\pi^2-0+C=-7$ or, $C=-7-\pi^2$
Hence, $v=4t^2-\cot t -7-\pi^2$
