Skip to main content
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 7.3.50
Chapter 7, Problem 7.3.50

37–56. Integrals Evaluate each integral.
∫ dx/x√(16 + x²)

Verified step by step guidance
1
Recognize that the integral is of the form \(\int \frac{dx}{x \sqrt{16 + x^2}}\), which suggests a trigonometric substitution because of the \(\sqrt{a^2 + x^2}\) term, where \(a = 4\).
Use the substitution \(x = 4 \tan(\theta)\), which implies \(dx = 4 \sec^2(\theta) d\theta\). This substitution simplifies the square root because \(\sqrt{16 + x^2} = \sqrt{16 + 16 \tan^2(\theta)} = 4 \sec(\theta)\).
Rewrite the integral in terms of \(\theta\) by substituting \(x\) and \(dx\) and simplifying the expression inside the integral.
Simplify the resulting integral to a form involving trigonometric functions that can be integrated more easily, such as \(\int \frac{4 \sec^2(\theta) d\theta}{4 \tan(\theta) \cdot 4 \sec(\theta)}\).
Integrate the simplified trigonometric integral, then substitute back \(\theta = \tan^{-1}(x/4)\) to express the answer in terms of \(x\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
7m

Key Concepts

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

Integration involving square roots of quadratic expressions

Integrals containing expressions like √(a² + x²) often require special techniques such as trigonometric substitution to simplify the integrand. Recognizing the form helps in choosing the appropriate substitution to transform the integral into a more manageable form.
Recommended video:
03:33
Integrals Involving Natural Logs: Substitution Example 7

Trigonometric substitution

Trigonometric substitution replaces variables with trigonometric functions to simplify integrals involving √(a² + x²), √(a² - x²), or √(x² - a²). For √(a² + x²), substituting x = a tan θ converts the square root into a secant function, facilitating easier integration.
Recommended video:
Guided course
6:04
Introduction to Trigonometric Functions

Integration of rational functions after substitution

After substitution, the integral often reduces to a rational function of trigonometric expressions. Understanding how to integrate these resulting functions, such as sec θ or tan θ, is essential to complete the integration and then revert back to the original variable.
Recommended video:
07:01
Integrals Involving Natural Logs: Substitution
Related Practice
Textbook Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.


lim x → 0⁺ (tanh x)ˣ

Textbook Question

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)

Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫ (x²) / (4x³ + 7) dx

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

22–36. Derivatives Find the derivatives of the following functions.


f(x) = x sinh⁻¹ x − √(x² + 1)

Textbook Question

37–56. Integrals Evaluate each integral.


∫ sech² w tanh w dw

1
views