Solution to Linear First Order Ordinary Differential Equation/Proof 2

Proof
From the Product Rule for Derivatives:

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

Integrating $x$ now gives:
 * $\displaystyle e^{\int P \left({x}\right) \, \mathrm d x} y = \int Q \left({x}\right) e^{\int P \left({x}\right) \, \mathrm d x} \, \mathrm d x + C$

whence we get the result by dividing by $e^{\int P \left({x}\right) \, \mathrm d x}$.