I am supposed to calculate $\int_{\gamma}\sin(2z)dz $ where $\gamma$ is the line segment joining $i+1$ to $-i$
Can we apply the fundamental theorem of calculus (because I think we are within the framework) and say the result is $\frac{-\cos(2z)}{2}$ evaluated between $i+1$ and $-i$, which gives us, $$\frac{-\cos(2+2i) + \cos(-2i)}{2}$$
Yes, you can. If $U$ is a domain in the complex plane, and $F:U\to\mathbb{C}$ is a holomorphic function, then $$\int_\gamma F'(z)\,dz = F(b)-F(a)$$ for any curve $\gamma$ that begins at $a$ and ends in $b$.
In your case, using the above $F(z)=-\cos(2z)/2$ is the most efficient way to get the result, although direct calculation with a parametrized segment is not hard, either.