Integrating Factor for First Order ODE/Conclusion

Theorem
Let the first order ordinary differential equation:
 * $(1): \quad M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

be non-homogeneous and not exact.

Let $\mu \left({x, y}\right)$be an integrating factor for $(1)$.

If one of these is the case:
 * $\mu$ is a function of $x$ only
 * $\mu$ is a function of $y$ only
 * $\mu$ is a function of $x + y$
 * $\mu$ is a function of $x y$

then:
 * $\mu = e^{\int f \left({w}\right) \mathrm d w}$

where $w$ depends on the nature of $\mu$.

Proof
We have one of these:
 * Integrating Factor for First Order ODE: Function of One Variable: $x$ or $y$ only
 * Integrating Factor for First Order ODE: Function of $x + y$
 * Integrating Factor for First Order ODE: Function of $x y$

We have an equation of the form:
 * $\dfrac 1 \mu \dfrac {\mathrm d \mu} {\mathrm d w} = f \left({w}\right)$

which is what you get when you apply the Chain Rule and Derivative of Logarithm Function to:
 * $\dfrac {\mathrm d \left({\ln \mu}\right)}{\mathrm d w} = f \left({w}\right)$

Thus:
 * $\displaystyle \ln \mu = \int f \left({w}\right) \mathrm d w$

and so:
 * $\mu = e^{\int f \left({w}\right) \mathrm d w}$

Hence the results as stated.