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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.5
Chapter 7, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Intro to Rational Functions

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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Completing the Square to Rewrite the Integrand

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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Euler's Method
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The variable y = t + 1 doubles in value whenever t increases by 1 unit.

Textbook Question

Radioactive decay The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?

Textbook Question

2–9. Integrals Evaluate the following integrals.


∫ dx / √(x² − 9),x > 3

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Textbook Question

2–9. Integrals Evaluate the following integrals.


∫ (eˣ / (4eˣ + 6)) dx

Textbook Question

2–9. Integrals Evaluate the following integrals.


∫₀¹ (x² / (9 − x⁶)) dx

Textbook Question

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.


a. cosh 0