While studying the Green's theorem, I think a lot about whether there is an abundance of the condition (now called $X$) of
$X:\quad$ $P$ and $Q$ (in the Wikipedia article, $L$ and $M$) have continuous, first-order partial derivatives.
In the proof in simple cases of the theorem (in the Wikipedia article), it seems that we only need
$X':\quad$ $P$ has the integrable $y$-partial derivative, and $Q$ has the integrable $x$-partial derivative.
So, my question: Can the condition $X$ of the Green's theorem be extended to $X'$ while the theorem still holds true?
Obviously, as my explanation of the posing process points out, the answer of my question may be lay on the non-"simple" cases of the theorem or the more-general theorem of Stokes. Either this or that, please elaborate it simply to me. In addition, I haven't study the Stokes' theorem yet.
Thanks in advance!