Split [tex]C[/tex] into two component segments, [tex]C_1[/tex] and [tex]C_2[/tex], parameterized by
[tex]\mathbf r_1(t)=(1-t)(0,0)+t(6,1)=(6t,t)[/tex]
[tex]\mathbf r_2(t)=(1-t)(6,1)+t(7,0)=(6+t,1-t)[/tex]
respectively, with [tex]0\le t\le1[/tex], where [tex]\mathbf r_i(t)=(x(t),y(t))[/tex].
We have
[tex]\mathrm d\mathbf r_1=(6,1)\,\mathrm dt[/tex]
[tex]\mathrm d\mathbf r_2=(1,-1)\,\mathrm dt[/tex]
where [tex]\mathrm d\mathbf r_i=\left(\dfrac{\mathrm dx}{\mathrm dt},\dfrac{\mathrm dy}{\mathrm dt}\right)\,\mathrm dt[/tex]
so the line integral becomes
[tex]\displaystyle\int_C(x+6y)\,\mathrm dx+x^2\,\mathrm dy=\left\{\int_{C_1}+\int_{C_2}\right\}(x+6y,x^2)\cdot(\mathrm dx,\mathrm dy)[/tex]
[tex]=\displaystyle\int_0^1(6t+6t,(6t)^2)\cdot(6,1)\,\mathrm dt+\int_0^1((6+t)+6(1-t),(6+t)^2)\cdot(1,-1)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(35t^2+55t-24)\,\mathrm dt=\frac{91}6[/tex]
