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 8.7.82
Chapter 8, Problem 8.7.82

79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
82. ∫ (sin⁻¹(ax)) / x² dx, a > 0

Verified step by step guidance
1
Identify the integral to solve: \(\int \frac{\sin^{-1}(ax)}{x^{2}} \, dx\), where \(a > 0\).
Consider using integration by parts, since the integrand is a product of functions involving an inverse trigonometric function and a power of \(x\). Let \(u = \sin^{-1}(ax)\) and \(dv = \frac{1}{x^{2}} dx\).
Compute the derivatives and integrals needed for integration by parts: find \(du = \frac{a}{\sqrt{1 - a^{2}x^{2}}} dx\) and \(v = -\frac{1}{x}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), substituting the expressions for \(u\), \(v\), and \(du\).
Simplify the resulting integral, which will likely require a second table look-up or substitution to evaluate \(\int \frac{1}{x \sqrt{1 - a^{2}x^{2}}} dx\). Use a table of integrals or a computer algebra system to find this integral.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions, like sin⁻¹(x), are the inverses of the standard trigonometric functions and return an angle whose sine is x. Understanding their properties and derivatives is essential for integrating expressions involving these functions.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It is useful when integrating products of functions, such as an inverse trigonometric function multiplied by an algebraic expression, by reducing the integral into simpler parts.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Use of Integral Tables and Computer Algebra Systems

Integral tables provide standard forms of integrals that can simplify complex integration problems. Computer algebra systems (CAS) help verify solutions and handle complicated integrals, ensuring accuracy and saving time.
Recommended video:
08:01
Integration Using Partial Fractions
Related Practice
Textbook Question

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

88. Evaluate ∫ dx/(2 + cos x).

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.

38. ∫ tan⁵θ sec⁴θ dθ

Textbook Question

94. The family f(x) = 1/xᵖ revisited Consider the family of functions f(x) = 1/xᵖ, where p is a real number.

For what values of p does the integral ∫(1 to ∞) 1/xᵖ dx exist?

What is its value when it exists?

1
views
Textbook Question

5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.

12. (2x² + 3)/((x² - 8x + 16)(x² + 3x + 4))

1
views
Textbook Question

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

23. ∫ 1/(25 - x²)^(3/2) dx