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The path of integration C consists of the straight lines say OA,AB,BC and CO where A,B,C are given respectively except the origin.LEt R be the plane region bounded by C.

Using Green's theorm,

\oint _{C}[(x^{2}-\cos hy)dx+(y+\sin x)dy]=\iint _{R}[{\frac  {\partial }{\partial x}}(y+\sin x)-{\frac  {\partial }{\partial y}}(x^{2}-\cos hy)]dxdy\,

=\int _{{x=0}}^{{\pi }}\int _{{y=0}}^{{1}}(\cos x+\sin hy)dxdy\,

=\int _{{0}}^{{\pi }}[y\cos x+\cos hy]_{{0}}^{{1}}dx\,

=\int _{{0}}^{{\pi }}(\cos x+\cos h1-1)dx=[\sin x+x\cos h1-x]_{{0}}^{{\pi }}\,

=\pi \cos h1-\pi =\pi (\cos h1-1)\,

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