1. **State the problem:** We need to solve the integral $$\int \left(\int (F_x + G_y)^2 \right)$$ where $F_x$ and $G_y$ denote partial derivatives of functions $F$ and $G$ with respect to $x$ and $y$ respectively. 2. **Recall the vector calculus identities:** Given $\nabla = (F_x + G_y) f^\perp$ and $\nabla = (F_y - G_x)$, these suggest relationships between partial derivatives of $F$ and $G$. 3. **Interpret the problem:** The inner integral is over some variable (likely $x$ or $y$), and the outer integral is over the other variable. Without explicit limits or variables, assume integration over $x$ then $y$. 4. **Express the integral:** $$\int \left( \int (F_x + G_y)^2 \, dx \right) dy$$ 5. **Expand the square:** $$ (F_x + G_y)^2 = F_x^2 + 2 F_x G_y + G_y^2 $$ 6. **Integrate term-by-term:** $$ \int (F_x^2 + 2 F_x G_y + G_y^2) \, dx = \int F_x^2 \, dx + 2 \int F_x G_y \, dx + \int G_y^2 \, dx $$ 7. **Note:** Since $G_y$ is a partial derivative with respect to $y$, it can be treated as constant with respect to $x$ integration, so: $$ \int F_x G_y \, dx = G_y \int F_x \, dx $$ 8. **Integrate $F_x$ with respect to $x$:** $$ \int F_x \, dx = F + C(y) $$ where $C(y)$ is an integration constant depending on $y$. 9. **Substitute back:** $$ \int (F_x + G_y)^2 \, dx = \int F_x^2 \, dx + 2 G_y (F + C(y)) + G_y^2 x + D(y) $$ where $D(y)$ is another integration constant. 10. **Now integrate with respect to $y$:** $$ \int \left[ \int (F_x + G_y)^2 \, dx \right] dy = \int \left( \int F_x^2 \, dx + 2 G_y F + 2 G_y C(y) + G_y^2 x + D(y) \right) dy $$ 11. **Without explicit forms of $F$ and $G$, the integral cannot be simplified further.** **Final answer:** The integral evaluates to $$ \int \left( \int (F_x + G_y)^2 \, dx \right) dy = \int \left( \int F_x^2 \, dx + 2 G_y F + 2 G_y C(y) + G_y^2 x + D(y) \right) dy $$ where $C(y)$ and $D(y)$ are functions of $y$ arising from integration constants. This is the most explicit form possible without additional information about $F$ and $G$.