Contour integral, f(z)=$ze^{z^2}$

Question

For part $(a)$ is the answer just $0$? Using Cauchy-Goursat theorem?

For part $(b)$ I am confused. Do I use

?

It seems very complicated. Am I missing a trick?

Answer 1

For a) apply the Cauchy's integral theorem. It is $0$.

For b) note that $\gamma(0)=0$ and $\gamma(1)=i$ so you can change $\gamma$ for any curve that begins at $0$ and ends at $i$, for example, $\gamma(t)=it$, $t\in[0,1]$.

Answer 2

I imagine since $f$ is entire (a) is $0$, (b) the path goes from $0$ to $i$ so instead of integrate on $\Gamma$, integrate on the straight line from $0$ to $i$.