How to compute the line integral of \int_{C^+}(x^2-y)dx+(y^2+x)dy?

I need help with this problem:

Compute the line integral$ \int_{C^+}(x^2-y)dx+(y^2+x)dy$ where $ C^+$ is the parabollic arc $ y=x^2+1$ , $ 0\leq x\leq 1$ oriented from $ (0,1)$ to $ (1,2)$ .

First I parametrized the arc $ C^+$ by setting $ x=t$ , then $ y=t^2+1$ , so the parametrization would be $ \alpha(t)=(t,t^2+1), t\in[0,1]$ . then I tried to solve this by using the formula $ \int_Cfds=\int_\alpha fds=\int_{0}^1f(\alpha(t))\Vert\alpha'(t)\Vert dt+\int_1^2 f(\alpha(t))\Vert\alpha'(t)\Vert dt$ . But I don’t know how to compute that, since I have an integral with respect to $ x$ and one with respect to $ y$ .