$$F(x,y)=\langle 2y^2+y^2\cos(xy^2),4xy+2xy\cos(xy^2)\rangle $$ $$r(t)=\langle e^t\sin(t),e^t\cos(t)\rangle,\quad t\in[0,\pi].$$
I found $$r'(t)= \langle \left[ e^t\sin(t) + e^t\cos(t) \right] \, dt, \left[ e^t \cos(t) - e^t\sin(t)\right] \, dt \rangle .$$
Now I know $\int_0^\pi F\cdot dr$, so I thought I substitute $e^t\sin(t)$ for $x$ and $e^t\cos(t)$ for $y$ and multiply by their respective $r(t)$ value, but the integral gets lengthy and complex very quickly. Is there something I'm missing?
They gave you a nightmarish integral on purpose. You were supposed to notice that $F$ is conservative. Then $F=\nabla f$ for some function $f$ and $$ \int_C F\cdot d\vec r=f(\vec r(\pi))-f(\vec r(0)). $$