I am trying to compute the line integral of the vector field <$2xy^2 + 3xz^2$, $2x^2y + 2y$, $3x^2z - 2z$>, on the curve C given by $\vec r(t)$ = ($\cos(2t)+5\sin(5t)$, $6\sin(t) + 4\sin(5t)$, $\cos(2t)+\cos(5t)$), for $0 < t < \pi$.
I get that I am supposed to differentiate $\vec r(t)$ and find the dot product of that and the vector field expressed in terms of t, then integrate that. However when I try to integrate I end up with a huge mess of sin and cosine terms and I don't know how to integrate that mess. Does anyone know if there is a more efficient way of dealing with the sin and cosine terms?
Any help is much appreciated!
Fortunately, your vector field is the gradient of $$\phi(x,y,z) = x^2y^2 + \frac{3}{2}x^2z^2 + y^2 - z^2$$
which, by the fundamental theorem of line integrals, implies that your integral is $\phi(\vec{r}(\pi)) - \phi(\vec{r}(0))$.