Calculate $$\int_\gamma \sin^2(z)\cos(z) dz,$$ where $\gamma$ is the path from $3\pi$ to $i$ consisting of the following two pieces: The bottom half of the circle $|z-2\pi| = \pi$ followed by the line segment from $3\pi$ to $i$.
I can't seem to get the right answer. I'm guessing I'm going wrong with parameterize $\gamma$.
My attempt at the parameterization:
$\gamma=\gamma_1 + \gamma_2$ where $$\gamma_1=3\pi e^{-it}, t \in [\pi,2\pi]$$ $$\gamma_2=-3\pi t+it+3\pi, t \in [0,1]$$ Thank you!
$$ \sin^2(z)\,\cos(z)=\left(\frac{1}{3}\sin^3(z)\right)' $$ and hence $$ \int_\gamma \sin^2(z)\,\cos(z)\,dz=\int_{3\pi}^i \sin^2(z)\,\cos(z)\,dz =\frac{1}{3}\sin^3(i)-\frac{1}{3}\sin^3(3\pi)=\cdots $$