Line integral $\int_C(3x + \sin(y)) ds$ where $C$ is line segment from $(1,2)$ to $(5,4)$

Question

How to calculate the line integral $$ \int_C(3x + \sin(y)) \,ds $$
where $C$ is line segment from $(1,2)$ to $(5,4)$?

Progress

I know $r(t) = \langle 1+4t,2+2t\rangle$ from $(0,1)$ so I have an integral from 0 to 1 of (???) dt

How do I take my original integral and parametrization to make a new integral?

I know $ds = \sqrt{(dx/ds)^2 + (dy/ds)^2}$

Answer 1

First, your $ds = \sqrt{(dx/dt)^2 + (dy/dt)^2}\ dt$. Then as you have already found \begin{align} x(t) &= 1 + 4t\\ y(t) &= 2 + 2t \end{align} What is $dx/dt$ and $dy/dt$? Once you have that you can plug what you know into your integral $$ \int_0^1[3x(t) + \sin(y(t))]ds = 40.7803 $$