Problem 1610 (difficulty: 3/10)

For \(\displaystyle a,r>0\) find the following integrals:

\(\displaystyle \frac1{2\pi i}\int\limits_{|z|=r}a^z\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z+1}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{z^2}\dz; \quad \frac1{2\pi i}\int\limits_{|z|=r}\frac{a^z}{(z+2)^2}\dz. \)

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