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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.RE.5
Capitolo 7, Problema 7.RE.5

2–9. Integrals Evaluate the following integrals.


∫ (x + 4) / (x² + 8x + 25) dx

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Start by examining the integral \( \int \frac{x + 4}{x^{2} + 8x + 25} \, dx \). Notice that the denominator is a quadratic expression, so consider completing the square to simplify it.
Rewrite the denominator \( x^{2} + 8x + 25 \) by completing the square: \( x^{2} + 8x + 25 = (x + 4)^{2} + 9 \). This form will help in recognizing the integral structure.
Split the integral into two parts by expressing the numerator in terms of \( (x + 4) \): \( \int \frac{x + 4}{(x + 4)^{2} + 9} \, dx \). This suggests a substitution where \( u = x + 4 \).
Use the substitution \( u = x + 4 \), so \( du = dx \). The integral becomes \( \int \frac{u}{u^{2} + 9} \, du \). This integral can be approached by recognizing it as a rational function where the numerator is the derivative of the denominator's inner function.
To solve \( \int \frac{u}{u^{2} + 9} \, du \), consider using the substitution \( w = u^{2} + 9 \), so \( dw = 2u \, du \). Rewrite the integral accordingly and split it if necessary to integrate using logarithmic and arctangent functions.

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Integration of Rational Functions

This involves integrating functions expressed as the ratio of two polynomials. Techniques often include polynomial division, substitution, or rewriting the integrand to simplify the integral into a more manageable form.
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Completing the Square

Completing the square transforms a quadratic expression into the form (x + a)² + b, which is useful for recognizing standard integral forms, especially those involving inverse trigonometric functions or logarithms.
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Substitution Method

Substitution simplifies integrals by changing variables to reduce the integral into a basic form. For example, setting u = x + constant or u = quadratic expression helps in integrating complex rational functions.
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2–9. Integrals Evaluate the following integrals.


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