Integrating Factors Found by Inspection

This section will use the following four exact differentials that occurs frequently.

1. $d(xy) = x \, dy + y \, dx$
    
2. $d\left( \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{y^2}$
    
3. $d\left( \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2}$
    
4. $d\left( \arctan \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2 + y^2}$

 

5. $d\left( \arctan \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{x^2 + y^2}$