CVXI5

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

We have dz=(1+2ti)dt,x=t,y=t^{2}\,.

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

=\int _{0}^{1}t^{3}+2it^{4}+it^{2}-2t^{3}\,dt={\frac  {-1}{4}}+i{\frac  {11}{15}}\,


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