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\frac{dy}{dx}+P(x)y=Q(x)

Multiply both sides by e^{\int P(x)dx} :
\frac{dy}{dx}e^{\int P(x)dx}+P(x)ye^{\int P(x)dx}=Q(x)e^{\int P(x)dx}

Use product rule on left hand side:
\frac{dy}{dx}e^{\int P(x)dx}+P(x)ye^{\int P(x)dx}=\frac{d}{dx}(ye^{\int P(x)dx})

Thus integrating on both sides

\frac{d}{dx}(ye^{\int P(x)dx})=Q(x) e^{\int P(x)dx}+C.

Delete the last C in the above post (typo), integrating

\frac{d}{dx}(ye^{\int P(x)dx})=Q(x)e^{\int P(x)dx}

leads to

y e^{\int P(x)dx}=\int Q(x)e^{\int P(x)dx} dx +C