how do I resolve this type of line integral?

Question

$r(t) = (2t\sqrt 3,1+2sin(t),2cos(t))$ with $t\in[0,2\pi]$

I need to solve : $\int_rxyz ds$

also I need to know how to resolve this type :

$\int_r F dr$

knowing that $F(x,y) = ({y\over1+xy},{x\over1+xy})$ and $r(t) = (cos(t),2sin(t))$ with $t\in[0,2\pi]$

Answer 1

If $f$ is a scalar function then:

$$\int_C f \ ds = \int_{a}^b f(r) \|r'\| \ dt$$

If $F$ is a vector field we have:

$$\int_{C} F \cdot d\textbf{r} = \int_{a}^b F(\gamma) \cdot \gamma' \ dt$$