Answer
$v=\dfrac{1}{2 \cos t} +\dfrac{1}{2}$
Work Step by Step
We need to find the anti-derivative for $\dfrac{dv}{dt}= \dfrac{1}{2} \sec t \tan t $
Thus, we have $v=\dfrac{1}{2 \cos t} +C$
Apply the initial condition $v(0)=1$ in the above equation to solve for $C$.
we get $\dfrac{1}{2}+C=1 \implies C=\dfrac{1}{2}$
Hence, $v=\dfrac{1}{2 \cos t} +\dfrac{1}{2}$