I am trying to work the following question and the approach I have been taught doesn't seem to work as it leads to some odd substitutions. I can't find an example similar to this anywhere.
Question as given:
Evaluate the line integral $\int_{C}cos(z)dx + e^{x}dy + e^{y}dz$ where $C$ is the curve parameterized as $C(t)=(1,t,e^{t}), 0 \leq t \leq 2$.
I have tried substituting in the parameterization as I have been taught, but I'm not sure how to go about substituting $dx, dy$ and $dz$ to obtain a workable equation? Apologies if I've missed something obvious here.
The parameterization carries all the information you need: $C(t)=(x(t),y(t),z(t))$, so the $x$-coordinate of all points on $C$ are $1$; the $y$-coordinate is equal to the value of the parameter $t$; and the $z$-coordinate is the value of $e^t$, where $0\le t\le2$.
Then the differentials of the components of $C(t)$ are
$$\begin{cases}\mathrm dx=0\\\mathrm dy=\mathrm dt\\\mathrm dz=e^t\,\mathrm dt\end{cases}$$
Substitute these, as well as $x(t),y(t),z(t)$, into the integral, and the rest is simple:
$$\begin{align*} \int_C\cos z\,\mathrm dx+e^x\,\mathrm dy+e^y\,\mathrm dz&=\int_0^2\cos(e^t)\cdot0+e^1\cdot\mathrm dt+e^t\cdot e^t\,\mathrm dt\\[1ex] &=\int_0^2e+e^{2t}\,\mathrm dt \end{align*}$$