The value of integral ${\int }_0^{1+\mathrm{i}}\mathrm{z} d\mathrm{z}$ along the line $\mathrm{y}=\mathrm{x}$ is _____ .

The value of integral ${\int }_{\mathrm{C}} \overline{\mathrm{z}} d\mathrm{z}$ , where C is shortest path joining A(0, 0) to B(1, 1) , is _____ .

The value of integral ${\int }_{\mathrm{C}} \frac{3\mathrm{z}}{\mathrm{z}+2} d\mathrm{z}$ , where $\mathrm{C} : \left|\mathrm{z}\right|=1$ , is _____ .

The value of integral ${\int }_{\mathrm{C}} \frac{\mathrm{dz}}{\mathrm{z}(\mathrm{z}-2)}$ , where $\mathrm{C} : \left|\mathrm{z}\right|=1$ , is _____ .

Let C be a circle $\left|\mathrm{z}-1\right|=3$ in the complex plane, then ${\int }_{\mathrm{C}} \frac{\cos \mathrm{z}}{\mathrm{z}-\mathrm{\pi }} d\mathrm{z}$ , is _____ .

As per cauchy's integral formula ${\int }_{\mathrm{C}} \frac{\mathrm{f}\left(\mathrm{z}\right)}{\mathrm{z}-{\mathrm{z}}_0} d\mathrm{z}=$ _____ .

As per cauchy's integral formula ${\int }_{\mathrm{C}} \frac{\mathrm{f}\left(\mathrm{z}\right)}{{(\mathrm{z}-{\mathrm{z}}_0)}^{\mathrm{n}+1}} d\mathrm{z}=$ _____ .

The value of ${\int }_{\mathrm{C}} \frac{{\mathrm{e}}^{\mathrm{z}}}{{(\mathrm{z}-1)}^2} d\mathrm{z}$ , $\mathrm{C} : \left|\mathrm{z}\right|=2$ , is _____ .

Let C be a circle $\left|z\right|=3$ in the complex plane & ${\int }_{\mathrm{C}}\frac{{\mathrm{e}}^{\mathrm{z}}}{{(\mathrm{z}-2)}^3}d\mathrm{z} = 2\mathrm{\pi ik}$ then k=_____ .

Let C be a circle $\mathit{\left|}\mathit{z}\mathit{\right|}\mathit{=}\mathit{3}$ in the complex plane & ${\int }_C\frac{\sin z}{{(z + \frac{\mathrm{\pi }}{2})}^3}dz = 2\mathrm{\pi ik}$ then k=_____ .