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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.56
Capitolo 7, Problema 7.3.56

37–56. Integrals Evaluate each integral.


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

Guida verificata passo dopo passo
1
Start by examining the integral \( \int \frac{dx}{x^{2} + 25x} \). Notice that the denominator can be factored as \( x(x + 25) \).
Rewrite the integral using the factorization: \( \int \frac{dx}{x(x + 25)} \). This suggests using partial fraction decomposition to simplify the integrand.
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 \( B \).
Solve for \( A \) and \( B \) by substituting convenient values for \( x \) (such as \( x = 0 \) and \( x = -25 \)) or by equating coefficients.
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.

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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.
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Intro to Rational Functions

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.
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Partial Fraction Decomposition: Distinct Linear Factors

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.
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Substitution With an Extra Variable
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