# Integrating Factors Found by Inspection

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

- $d(xy) = x \, dy + y \, dx$

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

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

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