Evaluating a line integral through a vector field in 3 dimensions.

Question

Let $\mathbf{F}(x,y,z) = (2xyz + \sin x)\mathbf{i} + (x^2 z)\mathbf{j} + (x^2 y)\mathbf{k}$.

Evaluate the integral of $\mathbf{F}$ along $c$, where $c(t) = (cos^5(t), sin^3(t), t^4)$, $t \in [0, 2\pi)$.

How do I start evaluating this integral? What knowledge and theorems should I know to be able to solve this? I know how to solve a line integral in $2$ dimensions and in 2D vector fields, but I have never tried with 3D. Thank you.

edit:

So would this be valid?

$$\int_{0}^{2\pi} -(2xyz + sinx)5cos^4(t)sin(t) + x^2z(3sin^2(t) cos(t) + 3x^2yt^3$$

Then just substitute all the x's with $cos^5(t)$ y's with $sin^3(t)$ and z's with $t^4$?

And just evaluate the integral?

Answer 1

Hint: We have and equivalent integral: $$\int_{0}^{2\pi} Mdx+Ndy+Pdz$$ where $M=(2xyz + \sin x)$, $N=x^2z$ and $P=x^2y$. And $x=\cos^5 t$, $y=\sin^3t$, $z=t^4$. Then you just have to compute $dx,dy,dx$ and replace in the integral. And of course remember to change $M,N,P$ in terms of $t$ this give us and ordinary integral.