(I was wondering how to use the given hint in this question.)
Evaluate $$ \int_L \frac{1}{x^2+y^2}\, ds $$
where $L$ is the straight line $Ax+By=C$, $C\ne 0$.
Hint: use the symmetry of the integrand to replace the line with a line having a simpler equation but giving the same value to the integral.
The symmetry of $x^2+y^2$ under rotation means $$ \int_L\frac{1}{x^2+y^2}\,\mathrm{d}s=\int_{L'}\frac{1}{x^2+y^2}\,\mathrm{d}s $$ where $L'$ is any other line with the same perpendicular distance from $0$. In particular, you may choose (say) the horizontal line $y=c$, where $c=C/\sqrt{A^2+B^2}$. So $$ \int_L\frac{1}{x^2+y^2}\,\mathrm{d}s=\int_{-\infty}^\infty\frac{1}{x^2+c^2}\,\mathrm{d}x $$ which is easy to calculate.