23-64. Integration Evaluate the following integrals.
57. ∫ (x³ + 5x)/(x² + 3)² dx

Verified step by step guidance

1. Start by examining the integral \( \int \frac{x^{3} + 5x}{(x^{2} + 3)^{2}} \, dx \). Notice that the denominator is \( (x^{2} + 3)^{2} \), which suggests a substitution involving \( x^{2} + 3 \) might simplify the integral.
2. Let \( u = x^{2} + 3 \). Then, compute the differential \( du = 2x \, dx \), which implies \( x \, dx = \frac{du}{2} \). This substitution will help rewrite parts of the integral in terms of \( u \) and \( du \).
3. Rewrite the numerator \( x^{3} + 5x \) as \( x(x^{2} + 5) \). Using the substitution, express \( x^{3} + 5x = x(x^{2} + 5) = x(u - 3 + 5) = x(u + 2) \). This allows you to write the integral as \( \int \frac{x(u + 2)}{u^{2}} \, dx \).
4. Replace \( x \, dx \) with \( \frac{du}{2} \) from the substitution step, so the integral becomes \( \int \frac{u + 2}{u^{2}} \cdot \frac{du}{2} = \frac{1}{2} \int \frac{u + 2}{u^{2}} \, du \).
5. Split the integral into simpler terms: \( \frac{1}{2} \int \left( \frac{u}{u^{2}} + \frac{2}{u^{2}} \right) du = \frac{1}{2} \int \left( \frac{1}{u} + 2u^{-2} \right) du \). Now, integrate each term separately using the power rule and logarithmic integration.

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include polynomial division, substitution, or partial fraction decomposition to simplify the integral into manageable parts.

Substitution Method

A technique where a part of the integral is replaced with a new variable to simplify the expression. For example, setting u = denominator or a function inside the integral can transform the integral into a standard form.

Polynomial Division and Simplification

When the degree of the numerator is equal to or higher than the denominator, dividing polynomials can simplify the integrand. This step often precedes substitution or partial fractions to make integration straightforward.