Use The Fundamental Theorem of Contour Integration or otherwise to evaluate the following integrals. (If it can't be used state why)

Question

a) $\int|z|dz$, where $\gamma(t) = 3e^{it} (0 \le t \le \pi)$;

b) $\int \cos z - z\sin z dz$, where $\gamma(t) = (−1 + 2t) + it (0 \le t \le 1).$

Answer 1

Some highlights:

$$z=3e^{it}\implies |z|=3\,,\,\,dz=3ie^{it}\,dt\implies \int_\gamma |z|dz=\int_0^\pi 3\cdot3ie^{it}\,dt=\ldots$$

The other one is a segment of straight line joining the point $\;(-i,0)\,,\,\,(1,1)\;$ in the complex plane, or if you prefer the complex notation: the point $\;-1\,,\,\,1+i\;$ , so directly:

$$(z\cos z)'=\cos z-z\sin z\implies\left.\int_\gamma (\cos z-z\sin z)dz=z\cos z\right|_{-1}^{1+i}=\ldots$$