23-64. Integration Evaluate the following integrals.
50. ∫ 8(x² + 4)/[x(x² + 8)] dx

Verified step by step guidance

1. Start by simplifying the integrand: \( \frac{8(x^2 + 4)}{x(x^2 + 8)} \). Notice that the numerator and denominator share some polynomial terms, so try to rewrite the expression to make it easier to integrate.
2. Split the fraction into partial fractions or separate terms if possible. For example, express \( \frac{8(x^2 + 4)}{x(x^2 + 8)} \) as a sum of simpler fractions like \( \frac{A}{x} + \frac{Bx + C}{x^2 + 8} \), where \(A\), \(B\), and \(C\) are constants to be determined.
3. Multiply both sides of the equation by the denominator \( x(x^2 + 8) \) to clear the fractions and set up an equation to solve for \(A\), \(B\), and \(C\). Equate coefficients of corresponding powers of \(x\) on both sides to find these constants.
4. Once you have the constants, rewrite the integral as the sum of integrals of the simpler fractions: \( \int \frac{A}{x} dx + \int \frac{Bx + C}{x^2 + 8} dx \).
5. Integrate each term separately using standard integral formulas: \( \int \frac{1}{x} dx = \ln|x| + C \), and for \( \int \frac{Bx + C}{x^2 + a^2} dx \), use substitution or recognize it as a combination of logarithmic and arctangent integrals.

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include simplifying the integrand, performing polynomial division if necessary, and decomposing the function into simpler parts for easier integration.

Partial Fraction Decomposition

A method used to break down complex rational expressions into a sum of simpler fractions. This technique is essential when the denominator factors into linear or irreducible quadratic terms, allowing each simpler fraction to be integrated individually.

Algebraic Simplification Before Integration

Simplifying the integrand by factoring, canceling common terms, or rewriting expressions can make integration more straightforward. Recognizing opportunities to reduce complexity helps in applying integration techniques effectively.