In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds

Verified step by step guidance

1. Identify the integrand: \( \frac{s^{4} + 81}{s (s^{2} + 9)^{2}} \). Since the denominator has a linear factor \( s \) and a repeated quadratic factor \( (s^{2} + 9)^{2} \), set up the partial fraction decomposition accordingly.
2. Write the partial fraction decomposition as: \[ \frac{s^{4} + 81}{s (s^{2} + 9)^{2}} = \frac{A}{s} + \frac{B s + C}{s^{2} + 9} + \frac{D s + E}{(s^{2} + 9)^{2}} \] where \( A, B, C, D, E \) are constants to be determined.
3. Multiply both sides of the equation by the denominator \( s (s^{2} + 9)^{2} \) to clear the fractions, resulting in: \[ s^{4} + 81 = A (s^{2} + 9)^{2} + (B s + C) s (s^{2} + 9) + (D s + E) s \].
4. Expand the right-hand side and collect like terms in powers of \( s \). Then, equate the coefficients of corresponding powers of \( s \) on both sides to form a system of equations for \( A, B, C, D, E \).
5. Solve the system of equations to find the values of \( A, B, C, D, E \). Once found, rewrite the integrand as the sum of partial fractions and integrate each term separately using standard integral formulas.

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with simpler denominators, often linear or quadratic factors, raised to appropriate powers.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler terms via partial fractions. Once decomposed, each term can be integrated using standard formulas, such as logarithmic or inverse trigonometric integrals, depending on the denominator's form.

Handling Repeated Quadratic Factors

When the denominator contains repeated quadratic factors, the partial fraction decomposition includes terms with increasing powers of the quadratic factor in the denominator. Each term typically has a linear numerator, and integrating these requires careful algebraic manipulation and knowledge of integration techniques for quadratic denominators.