Math Help - A vector equation thingy

1. A vector equation thingy

Calculate $\int_{C}f(X).dX$ where $f(x,y)=(x^2+xy,y-x^2y)$ and C is parametrized by $x=t, y= \frac{1}{t} \; (1\leq t \leq 3)$

$x=t \Rightarrow \tfrac{\delta x}{\delta t} = 1$
$y= \tfrac{1}{t} \Rightarrow \tfrac{\delta y}{\delta t} = -\tfrac{1}{t^2}$
$\int_{C}f(X).dX = \int_{C} (x^2+xy).\partial x + (y-x^2y).\partial y = \int_{1}^{3} (t^2+1-\frac{1}{t^3}+\frac{1}{t}).\partial t$
integrating that makes for a unpleasant mess: $\frac{t^3}{3}+t+\frac{1}{2t^2}+ln(t)$ which is why I'm wondering if I'm doing this right.
$\int_{C}f(X(t))\cdot X'(t) dt$