Find the number $N\in \mathbb{N} $ for which the following inequality is true, if $\sigma(t)=|z|=1 $, with $0\leq t\leq \pi/2$ and $\left | \int_{1}^{i} e^{z^{-1}} \right | \leq N \frac{\pi}{2}$. Please help me!
I do this: There is a circumference $\sigma(t)$ and I walked on its first quadrant. Then $\sigma(t)=\cos(t)+i \sin (t)$. I want to use this formula $$\int\limits_{\sigma} z \, dz=\int_{a}^{b} f(\sigma(t))\cdot \sigma'(t) \, dt$$ then I calculate $\sigma'(t)=i \cos(t)-\sin(t)$. Finally
$$\int\limits_{\sigma} z \, dz=\int_{0}^{\pi/2} (\cos{t}+i\sin{t})(i\cos{t}-\sin{t}) dt=\int_{0}^{\pi/2} (i\cos^2{t}+i\sin^2{t}-2\cos{t}\sin{t}) \, dt. $$
It is correct??