Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.7.76
Chapter 8, Problem 8.7.76

76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.

Verified step by step guidance
1
Step 1: Recognize that the integral ∫ (dx / (x√(x⁴ − 1))) involves inverse trigonometric functions, and different forms of these functions can represent the same mathematical relationship.
Step 2: Understand that the expressions ½ cos⁻¹(√(x⁻⁴)), ½ cos⁻¹(x⁻²), and ½ tan⁻¹(√(x⁴ − 1)) are equivalent but expressed differently due to trigonometric identities and transformations.
Step 3: Recall key trigonometric identities, such as cos⁻¹(u) + sin⁻¹(u) = π/2 and tan⁻¹(u) = sin⁻¹(u) / cos(u), which allow conversion between inverse trigonometric functions.
Step 4: Verify that the argument transformations (e.g., √(x⁻⁴) = x⁻² and √(x⁴ − 1)) are consistent with the domain and range of the integral, ensuring that all forms are valid representations.
Step 5: Conclude that the apparent discrepancy arises from different ways of expressing the same integral result, and all three forms are mathematically correct due to the equivalence of trigonometric identities.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8m
Was this helpful?

Key Concepts

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

Integration and Antiderivatives

Integration is the process of finding the antiderivative of a function, which represents the area under the curve of that function. Different methods of integration, such as substitution or integration by parts, can yield different forms of the same antiderivative. This means that multiple expressions can represent the same integral, differing only by a constant or a change in variable.
Recommended video:
05:50
Antiderivatives

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ and tan⁻¹, are used to find angles when given a ratio of sides in a right triangle. These functions can have multiple representations due to their periodic nature and the different quadrants in which angles can lie. Understanding the relationships between these functions is crucial for recognizing how different forms of an expression can be equivalent.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions

Equivalence of Expressions

Mathematically, two expressions are considered equivalent if they yield the same value for all inputs in their domain. In the context of integrals, different forms can arise from algebraic manipulation or the application of trigonometric identities. Recognizing that these forms can be transformed into one another through valid mathematical operations is essential for resolving apparent discrepancies in results.
Recommended video:
6:36
Simplifying Trig Expressions
Related Practice
Textbook Question

60–69. Completing the square Evaluate the following integrals.

65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)

1
views
Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

11. ∫ sin²(3x) dx

Textbook Question

90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)

and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.

Textbook Question

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

29. ∫ e⁻ˣ sin(4x) dx