Solution to Linear First Order Ordinary Differential Equation/Proof 2

Proof
From the Product Rule for Derivatives:

Hence, multiplying $(1)$ all through by $e^{\int \map P x \rd x}$:
 * $\dfrac \d {\d x} \left({e^{\int \map P x \rd x} y}\right) = \map Q x e^{\int \map P x \rd x}$

Integrating $x$ now gives:
 * $\displaystyle e^{\int \map P x \rd x} y = \int \map Q x e^{\int \map P x \rd x} \rd x + C$

whence we get the result by dividing by $e^{\int \map P x \rd x}$.