Solving simple PDE using partial integral

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I have a question regarding these notes for solving simple PDE. Here is the image:

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After finding $P_1, Q_1, R_1$, they have not made it exact differential, as I was taught, but have partially integrated with respect to one variable treating others as constants. Is this correct?

Is this only valid when $PP_1 + QQ_1 + RR_1 = 0$ ?