I'm trying to solve the problem $\int_C x\sin(y)ds$, where C is the line segment from (0,1) to (4,4). This is my solution, which is wrong. I was hoping someone could tell me where I took a misstep.
Parametric equations: $x=4t$ and $y=1+3t$,
$\int_0^14t\sin(3t+1) \sqrt({4^2}+{3^2})dt$
$20\int_0^1t\sin(3t+1)dt$
after using integration by parts:
$20[\frac{-t}{3}\cos(1+3t)+\frac{1}{9}\sin(1+3t)]|^1_0$
which after simplifying gives:
$\frac{20}{9}[\sin(4)-3\cos(4)].$
Notice that in the last passage you have: $$20\left[-\frac{\cos({4})}{3}+\frac{\sin({4})-\sin({1})}{9}\right] = \frac{20}{9}\left[-3\cos({4})+\sin({4})-\sin({1})\right]$$