# Evaluating This Complex Line integral

I'm trying to evaluate the following:

Came out as a mess... I'm thinking I should be using Cauchy’s Theorem but I'm not sure if it applies here and how to prove f(z) is analytic in this case.

Thanks.

• You could try letting $z=\sin(t^2)-i\frac{2t^2}{\pi}$ and substitute into the integral. Use the limits $0$ and $\sqrt{\pi/2}$. But maybe you tried that already. – Pixel Aug 27 '15 at 13:28
• You mean into the complex line integral formula or straight into i.imgur.com/RGsn44F.png? It comes out as HUGE, is that reasonable? Sure, I'll do that from now on. – Joe Aug 27 '15 at 13:31
• Yes. But as you say it's rather ugly ! See Jack's answer below... – Pixel Aug 27 '15 at 13:34

$$f(z)=z^3 e^{-z^4}$$ is an entire function having $F(z)=-\frac{1}{4}e^{-z^4}$ has a primitive. That implies: $$\int_{\mathcal{C}} f(z)\,dz = F(b)-F(a),$$ where $a=0$ and $b=1-i$ are the endpoints of $\mathcal{C}$.