Let $F(x,y,z) = \left( 2xyz + \sin(x)\right)\mathbf i + x^2z \mathbf j +$ $x^2 y \mathbf k $. Evaluate $$ \int_C F \cdot ds$$ where C is the parametrized curve $ c(t) = \left(\cos^5(t),\sin^3(t),t^4\right), 0\le t \le \pi $.
I tried to calculate this line integral by calculating the derivative of the c and dot product of the F and c' as the integrand. But the calculation is so complicated, so I think there might be much easier way to do this.
Observe that
$$\text{curl}\,F:=\begin{vmatrix}i&j&\;\;k\\\frac\partial{\partial x}&\frac\partial{\partial y}&\;\;\frac\partial{\partial z}\\2xyz+\sin x&x^z&\;\;x^2y\end{vmatrix}=0$$