Correct way to calculate a complex integral? I have
$$
\int_{[-i,i]} \sin(z)\,dz
$$
Parametrizing the segment $[-i,i]$ I have, if $t\in[0,1]$
$$
z(t) = it + (1-t)(-i) = 2it-i, \quad \dot{z}(t) = 2i.
$$
So
$$
\int_{[-i,i]} \sin(z)\,dz = \int_0^1 \sin(2it-i)2i\, dt = -\cos(2it-i)|_0^1 = 0.
$$
Am I correct?

Sine is analytic function, so you can also use Cauchy's theorem. 
$$
\int_{[-i, i]} \sin z\ dz = -\left . \cos z \right |_{-i}^i = 0
$$
