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Question

# Evaluate the following integrals. $\int \frac { z + 1 } { z \left( z ^ { 2 } + 4 \right) } d z$

Solution

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$\textbf{ We multiply both sides of equation by simple factors }$

\begin{align*} \frac { z + 1 } { z \left( z ^ { 2 } + 4 \right) } &= \frac { A } { z } + \frac { B z + C } { z ^ { 2 } + 4 } \\ \frac { z + 1 } { z \left( z ^ { 2 } + 4 \right) }&= \frac { A \left( z ^ { 2 } + 4 \right) + ( B z + C ) z } { z \left( z ^ { 2 } + 4 \right) }\\ A \left( z ^ { 2 } + 4 \right) + ( B z + C ) z &= z + 1 \end{align*}

The goal is to find the undetermined coefficients A, B, and C

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