$\begin{aligned} & \text { We have, } x+y \frac{d y}{d x}=\sec \left(x^2+y^2\right) \\ & \text { Put } x^2+y^2=u \Rightarrow 2 x+2 y \frac{d y}{d x}=\frac{d u}{d x} \\ & \therefore x+y \frac{d y}{d x}=\frac{1}{2} \frac{d u}{d x} \\ & \therefore \frac{1}{2} \frac{d u}{d x}=\sec u \\ & \therefore \int \frac{d u}{\sec u}=\int 2 d x \\ & \therefore \sin u=2 x+c \\ & \Rightarrow \sin \left(x^2+y^2\right)=2 x+c\end{aligned}$