91Ó°ÊÓ

Use the residue theorem to evaluate $$ \int_{C}[\\{5 z-2\\} /\\{z(z-1)\\}] d z $$ where \(\mathrm{C}\) is the circle \(|\mathrm{z}|=2\) described counterclockwise.

Use the residue theorem to evaluate $$ \int_{C}\left[\left\\{\left(1+z^{5}\right) \sinh z\right\\} /\left\\{z^{6}\right\\}\right] d z $$ where \(\mathrm{C}\) is the unit circle \(|z|=1\) described in the positive sense (i.e., counterclockwise).

Evaluate \(\int_{C}\left[\left\\{z e^{z}\right\\} /\left\\{z^{2}-1\right\\}\right] d z\) where \(C\) is the circle \(|z|=2\) taken in the counterclockwise direction.

Find the partial fraction expansion of $$ f(z)=\left\\{\left(z^{2}+1\right) /\left(z^{3}+4 z^{3}+3 z\right)\right\\} $$ using the theory of residues.

Evaluate \(\mathrm{I}=2 \pi \int_{0}\\{1 /(\cos \theta+2)\\} \mathrm{d} \theta\)

Evaluate \(\mathrm{I}=\pi \int_{-\pi} \mathrm{e}^{2 \cos \theta} \mathrm{d} \theta\)

Evaluate \(\mathrm{I}_{0}={ }^{\infty} \int_{0}\left\\{\mathrm{~d} \mathrm{x} /\left(\mathrm{x}^{2}+1\right)\right\\}\)

Evaluate \(\mathrm{I}_{0}={ }^{\infty} \int_{0}\left[\left\\{\mathrm{x}^{2} \mathrm{~d} \mathrm{x}\right\\} /\left\\{\left(\mathrm{x}^{2}+9\right)\left(\mathrm{x}^{2}+4\right)^{2}\right\\}\right]\).

Evaluate \(I_{1}=^{\infty} \int_{-\infty}\left\\{(x \cos x) /\left(x^{2}+1\right)\right\\} \mathrm{d} x\) and \(I_{2}={ }^{\infty} \int_{-\infty}\left\\{(\mathrm{x} \sin \mathrm{x}) /\left(\mathrm{x}^{2}+1\right)\right\\} \mathrm{d} \mathrm{x}\).