Finding the value of $\int{F \cdot dr}$

Question

Find the value of $\int{F \cdot \mathrm{d}r}$, where $$F(x,y) = \langle 5e^y+ye^x,e^x+5xe^y \rangle$$ and $$C: r(t) = \left\langle\sin\left(\frac{\pi t}{2}\right),\ln(t)\right\rangle; 1\le t\le2$$ So far, I've tried substituting in using the parametriziation suggested by the $r(t)$ function into the $F(x,y)$ vector field but that produced such a big integral there's no way of evaluating it. Any help would be welcome, thank you in advance!

Answer 1

As seen in the comments (and corrected on one key point), we can parametrize and set up the integral $$\int_C F\cdot d\mathbf{r} = \int_1^2 [5t+\ln(t)e^{\sin(at)}]a\cos at + \frac{e^{\sin(at)}+5\sin(at)e^{\ln t}}{t}\,dt$$ where $a=\frac{\pi}{2}$, abbreviated to simplify the typography.

This works, but it's messy.

Instead, note that we can write $F$ as a gradient $F=\nabla G$, where $G(x,y)=5xe^y+ye^x$. Then $G$ acts as an antiderivative; we can apply the line integral version of the FTC to get $$\int_C F\cdot d\mathbf{r} = G(r(2))-G(r(1)) = G(0,\ln 2) - G(1,0) = \ln 2 - 5$$