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

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.67

Chapter 7, Problem 7.3.67

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁/₈¹ dx/x√(1 + x²/³)

Verified step by step guidance

First, rewrite the integral to clearly identify the integrand: \( \int_{\frac{1}{8}}^{1} \frac{1}{x \sqrt{1 + \frac{x^{2}}{3}}} \, dx \).

Simplify the expression inside the square root: \( \sqrt{1 + \frac{x^{2}}{3}} = \sqrt{\frac{3 + x^{2}}{3}} = \frac{\sqrt{3 + x^{2}}}{\sqrt{3}} \). Substitute this back into the integrand.

Rewrite the integrand as \( \frac{1}{x} \cdot \frac{\sqrt{3}}{\sqrt{3 + x^{2}}} = \frac{\sqrt{3}}{x \sqrt{3 + x^{2}}} \).

Consider a substitution to simplify the integral. Let \( t = \frac{\sqrt{3 + x^{2}}}{x} \). Then express \( dt \) in terms of \( dx \) and rewrite the integral in terms of \( t \).

Use Theorem 7.7, which relates to integrals of the form \( \int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C \), to express the integral in terms of logarithms after substitution and simplification.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

10m

Was this helpful?

Key Concepts

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

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Integration Techniques for Functions Involving Roots and Rational Expressions

Integrals involving expressions like 1/(x√(1 + x^(2/3))) often require substitution or algebraic manipulation to simplify the integrand. Recognizing patterns or rewriting the integrand can help transform it into a form suitable for standard integral formulas or logarithmic expressions.

Theorem 7.7 and Logarithmic Integration

Theorem 7.7 typically refers to a result that expresses certain integrals in terms of logarithms, often involving integrals of the form ∫(f'(x)/f(x)) dx = ln|f(x)| + C. Applying this theorem helps rewrite the integral's antiderivative using logarithmic functions, which is essential for the problem's requirement.

Recommended video:

06:11

Fundamental Theorem of Calculus Part 1

Related Practice

Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = ln(3 sin² 4x)

Textbook Question

37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

Textbook Question

Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 75) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?

Textbook Question

37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

views

Textbook Question

37–56. Integrals Evaluate each integral.

∫ tanh²x dx (Hint: Use an identity.)

views

Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)