\begin{align} \int_C f(x,y) ds &= \int_0^1 f(x,x) dx \\ &= \int_0^1 \frac{6x}{3x^2+1} dx \\ &= \left[ \log \left( 3x^2+1 \right) \right]_0^1 \\ &= 2 \log 2 \end{align}