23-64. Integration Evaluate the following integrals.
54. ∫ (z + 1)/[z(z² + 4)] dz

Verified step by step guidance

1. Start by expressing the integrand \( \frac{z + 1}{z(z^2 + 4)} \) as a sum of partial fractions. Since the denominator factors into \( z \) and \( z^2 + 4 \), set up the decomposition as \( \frac{z + 1}{z(z^2 + 4)} = \frac{A}{z} + \frac{Bz + C}{z^2 + 4} \).
2. Multiply both sides of the equation by the denominator \( z(z^2 + 4) \) to clear the fractions, resulting in \( z + 1 = A(z^2 + 4) + (Bz + C)z \).
3. Expand the right-hand side to get \( z + 1 = A z^2 + 4A + B z^2 + C z \), then group like terms: \( (A + B) z^2 + C z + 4A \).
4. Equate the coefficients of corresponding powers of \( z \) on both sides: For \( z^2 \), \( A + B = 0 \); for \( z \), \( C = 1 \); and for the constant term, \( 4A = 1 \). Solve this system to find \( A, B, \) and \( C \).
5. Rewrite the integral using the partial fractions found, then integrate each term separately: \( \int \frac{A}{z} dz + \int \frac{Bz + C}{z^2 + 4} dz \). Use standard integral formulas such as \( \int \frac{1}{z} dz = \ln|z| + C \) and for the second integral, consider splitting it into two integrals and using substitution or recognizing the form \( \int \frac{z}{z^2 + a^2} dz \) and \( \int \frac{1}{z^2 + a^2} dz \).

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. It involves expressing a rational function as a sum of simpler terms with linear or quadratic denominators, facilitating straightforward integration.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts, such as polynomials or partial fractions. Recognizing standard integral forms and applying appropriate methods like substitution or partial fractions is essential for solving these integrals.

Handling Quadratic Denominators

When the denominator includes irreducible quadratic factors, integration may involve logarithmic or inverse trigonometric functions. Understanding how to integrate terms with quadratic denominators, often after partial fraction decomposition, is crucial for evaluating such integrals.