VC5.10

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Using Green's theorm, we have

\oint _{C}[(y-\sin x)dx+\cos xdy]=\iint _{R}[{\frac  {\partial }{\partial x}}(\cos x)-{\frac  {\partial }{\partial y}}(y-\sin x)]dxdy\,

=\int _{{x=0}}^{{{\frac  {\pi }{2}}}}\int _{{y=0}}^{{{\frac  {2x}{\pi }}}}(-\sin x-1)dxdy\,

=\int _{{0}}{{\frac  {\pi }{2}}}[-y\sin x-y]_{{0}}^{{{\frac  {2x}{\pi }}}}dx\,

={\frac  {-2}{\pi }}\int _{{0}}^{{{\frac  {\pi }{2}}}}x\sin xdx-{\frac  {2}{\pi }}\int _{{0}}^{{{\frac  {\pi }{2}}}}xdx=-[{\frac  {\pi }{4}}+{\frac  {2}{\pi }}]\,

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