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 8.6.79
Chapter 8, Problem 8.6.79

7–84. Evaluate the following integrals.
79. ∫ (arcsinx)/x² dx

Verified step by step guidance
1
Step 1: Recognize that the integral involves the function arcsin(x) divided by x². This suggests that integration by parts might be a useful technique. Recall the formula for integration by parts: ∫u dv = uv - ∫v du.
Step 2: Choose u = arcsin(x) and dv = 1/x² dx. This choice is made because the derivative of arcsin(x) simplifies nicely, and 1/x² dx is straightforward to integrate.
Step 3: Compute du by differentiating u = arcsin(x). The derivative of arcsin(x) is 1/√(1-x²), so du = (1/√(1-x²)) dx.
Step 4: Compute v by integrating dv = 1/x² dx. The integral of 1/x² is -1/x, so v = -1/x.
Step 5: Substitute u, v, du, and dv into the integration by parts formula: ∫(arcsin(x)/x²) dx = uv - ∫v du. This becomes (-arcsin(x)/x) - ∫(-1/x)(1/√(1-x²)) dx. Simplify and proceed to evaluate the remaining integral.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. In this context, evaluating the integral of a function like (arcsin x)/x² requires knowledge of techniques such as substitution or integration by parts.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Arcsine Function

The arcsine function, denoted as arcsin(x), is the inverse of the sine function, providing the angle whose sine is x. It is defined for values of x in the range [-1, 1] and outputs angles in radians between -π/2 and π/2. Understanding the properties and behavior of the arcsine function is crucial when integrating expressions that involve it, as it can affect the choice of integration techniques.
Recommended video:
06:21
Properties of Functions

Improper Integrals

An improper integral is an integral that has either infinite limits of integration or an integrand that approaches infinity within the interval of integration. In the case of ∫ (arcsin x)/x² dx, it is important to determine if the integral converges or diverges, especially near points where x approaches 0 or infinity. Evaluating improper integrals often involves taking limits to ensure the integral is well-defined.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Related Practice
Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

28. ∫ (from 1 to ∞) tan⁻¹(s)/(s² + 1) ds

1
views
Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

56. ∫ (from 0 to 1) 1/(x + √x) dx

1
views
Textbook Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

16. ∫ (from -∞ to ∞) (1/(x² + a²)) dx, a > 0

1
views
Textbook Question

4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.

Textbook Question

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

58. ∫₀^{2π} dt / (4 + 2 sin t)²

Textbook Question

68. Different methods

a. Evaluate ∫(cot x csc² x) dx using the substitution u=cotx.