Solution
= Solution
The only denominator zero is $z_0=-1$, which lies strictly inside the given circle. The numerator is an <entire function>. Apply the <derivative> form of the <Cauchy integral formula> with $n=6$:
$$
\oint_C\frac{\sin z}{(z+1)^7}\,dz
=\frac{2\pi i}{6!}\left.\frac{d^6}{dz^6}\sin z\right|_{z=-1}.
$$
The sixth <derivative> is $-\sin z$, so its value at $-1$ is $\sin1$. Consequently
$$
\boxed{\oint_C\frac{\sin z}{(z+1)^7}\,dz=\frac{\pi i\sin1}{360}}.
$$