Evaluate the integrals in Exercises 33–36.
∫ [1 / (x(9 - x²))] dx

Verified step by step guidance

1. Start by recognizing that the integral involves a rational function with a quadratic expression in the denominator: \(\int \frac{1}{x(9 - x^{2})} \, dx\).
2. Use partial fraction decomposition to rewrite the integrand. Express \(\frac{1}{x(9 - x^{2})}\) as \(\frac{A}{x} + \frac{Bx + C}{9 - x^{2}}\).
3. Multiply both sides by the common denominator \(x(9 - x^{2})\) to get an equation without denominators: \(1 = A(9 - x^{2}) + (Bx + C)x\).
4. Expand and group like terms by powers of \(x\): \(1 = 9A - A x^{2} + B x^{2} + C x\).
5. Equate coefficients of corresponding powers of \(x\) on both sides to form a system of equations for \(A\), \(B\), and \(C\). Then solve this system to find the values of \(A\), \(B\), and \(C\).

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 the integrand as a sum of fractions with simpler denominators, often linear or quadratic, facilitating straightforward integration.

Integration of Rational Functions

Integrating rational functions often requires algebraic manipulation such as partial fractions or substitution. Recognizing the form of the integrand helps determine the appropriate method, enabling the evaluation of integrals involving polynomials in numerator and denominator.

Logarithmic Integration

Logarithmic integration arises when integrating functions of the form 1/u, where u is a differentiable function of x. The integral results in a natural logarithm, ln|u| + C, which is common when integrating terms obtained after partial fraction decomposition.