By Green's theorem,
[tex]\displaystyle\int_C7y\,\mathrm dx+6x\,\mathrm dy[/tex]
[tex]=\displaystyle\iint_{x^2+y^2\le1}\frac{\partial(6x)}{\partial x}-\frac{\partial(7y)}{\partial y}\,\mathrm dx\,\mathrm dy[/tex]
[tex]=-\displaystyle\iint_{x^2+y^2\le1}\mathrm dx\,\mathrm dy[/tex]
which is -1 times the area of the disk [tex]x^2+y^2\le1[/tex]. Recall the area of a circle with radius [tex]r[/tex] is [tex]\pi r^2[/tex], so the integral has a value of [tex]\boxed{-\pi}[/tex].
