CVXI4

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Evaluate \int _{C}(xy+ix^{2})dz,C=z(t)=t+it,0\leq t\leq 1\,.

We have dz=(1+i)dt\, and x=t,y=t\,.

The integral is

\int _{0}^{1}(t^{2}+it^{2})(1+i)dt\,

=\int _{0}^{1}t^{2}(1+i)^{2}\,dt=2i\int _{0}^{1}t^{2}dt={\frac  {2i}{3}}\,


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