Step-by-Step Solution
Key Concept: Check for exact ODE or integrating factor; note $x+x^3=x(1+x^2)$
$dx+(1+x^2)(y\,dx+x\,dy)=0\Rightarrow dx+(1+x^2)d(xy)=0$. Let $u=xy$: $\int\frac{dx}{1+x^2}+u=c... $ Actually $\frac{dx}{1}+(1+x^2)du/(1)=0$... $\frac{du}{dx}=-(1+x^2)^{-1}/(1)$. $u=-\tan^{-1}x+c$. $xy+\tan^{-1}x=c$.
Correct Answer: 1