VC5.2

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Here the path of integration C in the given integral consists of the coordinates (0,0),(\pi ,0),(\pi ,{\frac  {\pi }{2}}),(0,{\frac  {\pi }{2}})\,.

Let R be the plane region bounded by C.

Therefore,\int _{C}(e^{{-x}}\sin ydx+e^{{-x}}\cos ydy)\,

=\iint _{R}[{\frac  {\partial }{\partial x}}(e^{{-x}}\cos y)-{\frac  {\partial }{\partial y}}(e^{{-x}}\cos y)]dxdy\, by using Green's theorm.

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

=-2\int _{{x=0}}^{{\pi }}\int _{{y=0}}^{{{\frac  {\pi }{2}}}}e^{{-x}}\cos ydxdy\,

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

=-2\int _{{x=0}}^{{\pi }}e^{{-x}}dx=-2[-e^{{-x}}]_{{0}}^{{\pi }}=-2(-e^{{-\pi }}+1)=2(e^{{-\pi }}-1)\,

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