37–56. Integrals Evaluate each integral.


∫₂₅²²⁵ dx / (x² + 25x) (Hint: √(x² + 25x) = √x √(x + 25).)

Verified step by step guidance

1. Start by examining the integral \( \int \frac{dx}{x^{2} + 25x} \). Notice that the denominator can be factored as \( x(x + 25) \).
2. Rewrite the integral using the factorization: \( \int \frac{dx}{x(x + 25)} \). This suggests using partial fraction decomposition to simplify the integrand.
3. Set up the partial fractions: \( \frac{1}{x(x + 25)} = \frac{A}{x} + \frac{B}{x + 25} \). Multiply both sides by \( x(x + 25) \) to find \( A \) and \( B \).
4. Solve for \( A \) and \( B \) by substituting convenient values for \( x \) (such as \( x = 0 \) and \( x = -25 \)) or by equating coefficients.
5. Once you have \( A \) and \( B \), rewrite the integral as \( \int \left( \frac{A}{x} + \frac{B}{x + 25} \right) dx \) and integrate each term separately using the natural logarithm function.

Key Concepts

Integration of Rational Functions

This involves integrating functions expressed as ratios of polynomials. Techniques such as partial fraction decomposition are often used to simplify the integrand into manageable terms that can be integrated directly.

Partial Fraction Decomposition

A method to break down complex rational expressions into simpler fractions. It is especially useful when the denominator factors into linear or quadratic terms, allowing easier integration of each component.

Algebraic Manipulation and Substitution

Rewriting expressions, such as recognizing that √(x² + 25x) = √x √(x + 25), helps simplify the integral. Substitution techniques can then be applied to transform the integral into a standard form for easier evaluation.