Complex integration with paramaterisation

Question

$\int z\sin(z^2)dz $ where $\gamma(t):= \sqrt{3}t + 2(1-t)i,$ where $0<= t<=1$.
I found this question on a previous years problem sheet but have never seen a question of this format(using parametrisation), how is this different to a 'typical' Cauchy Integral question and how do you go about solving it?

Answer 1

Since $\gamma$ is not a loop, you can't use that theorem. Instead, use the fact that $-\frac12\cos(z^2)$ is a primitive of the function that you want to integrate.