Line integral along the boundary of a square

Question

Calculate $$\int_\gamma(\cos x+\cos y+e^{x^2})dx+(\sin x\sin y+(y^4+1)^{\frac{1}{4}})dy$$ where $\gamma$ is the boundary of the square $[0,\frac{\pi}{2}]\times [0,\frac{\pi}{2}]$ with positive direction.

Any help? I don't know how to to this.

Answer 1

Hint. By Green's theorem, we are able to convert the line integral around the boundary of the square into double integral over the square: $$I:=\int_\gamma(\cos x+\cos y+e^{x^2})dx+(\sin x\sin y+(y^4+1)^{\frac{1}{4}})dy\\ =\iint_{[0,\frac{\pi}{2}]\times [0,\frac{\pi}{2}]}\left(\frac{\partial (\sin x\sin y+(y^4+1)^{\frac{1}{4}})}{\partial x}-\frac{\partial (\cos x+\cos y+e^{x^2})}{\partial y}\right) dx dy$$ Note that after taking the partial derivatives, the integrand of the double integral is not so scaring: $$I=\iint_{[0,\frac{\pi}{2}]\times [0,\frac{\pi}{2}]}\left( \cos x \sin y+\sin y \right) dx dy$$ Can you take it from here?