Given question :
Evaluate
$$\int_C \left(z^2+1\right)^2\,\mathbb dz $$
along the arc of the cycloid $x=a(\theta -\sin \theta),\,y=a(1-\cos \theta)$ from the point where $\theta=0$ to the point where $\theta =2\pi$
What should i do?
Should i substitute $z=x+iy,\,\mathbb dz=\mathbb dx+i \mathbb dy$, then just replace $x=a(\theta -\sin \theta),\,y=a(1-\cos \theta)$, then find the derivative of them w.r.t. $\theta$, and the integration is evaluated from $0$ to $2\pi$?
Or should i consider the "arc" word, which is i have to use arc length integral formula? I'm not sure.
Thanks. Anyway just let me do this by myself. I just need a hint.
The function that you are integrating is $(z^2+1)^2$, which has a primitive: $F(z)=\frac{z^5}5+\frac{2z^3}3+z$. So,$$\int_C(z^2+1)^2\,\mathrm dz=F(2\pi a)-F(0).$$