Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.3.3

Chapter 8, Problem 8.3.3

3. Describe the method used to integrate sin³x.

Verified step by step guidance

To integrate \( \sin^3(x) \), start by recognizing that this is a power of sine function. A common approach is to use a trigonometric identity to simplify the expression. Specifically, split \( \sin^3(x) \) into \( \sin(x) \cdot \sin^2(x) \).

Next, use the Pythagorean identity \( \sin^2(x) = 1 - \cos^2(x) \) to rewrite \( \sin^2(x) \). This gives \( \sin^3(x) = \sin(x) \cdot (1 - \cos^2(x)) \).

Now, let \( u = \cos(x) \). Then, \( du = -\sin(x) \, dx \). Substitute \( \sin(x) \, dx \) with \( -du \) and \( \cos(x) \) with \( u \). The integral becomes \( -\int (1 - u^2) \, du \).

Break the integral into two simpler parts: \( -\int 1 \, du + \int u^2 \, du \). Integrate each term separately using basic power rule for integration.

Finally, after integrating, substitute back \( u = \cos(x) \) to return to the original variable \( x \). Simplify the result to complete the solution.

Verified video answer for a similar problem:

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

Video duration:

Was this helpful?

Key Concepts

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

Integration Techniques

Integration techniques are methods used to find the integral of a function. For sin³x, one common technique is to use trigonometric identities and substitution to simplify the integrand. Understanding these techniques is essential for solving integrals that involve powers of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. For sin³x, the identity sin²x = 1 - cos²x can be used to rewrite the integrand, making it easier to integrate. Familiarity with these identities is crucial for manipulating trigonometric expressions in integration.

Substitution Method

The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. In the case of sin³x, substituting u = cosx can transform the integral into a more manageable form. This method is particularly useful when dealing with composite functions or powers of trigonometric functions.

Recommended video:

07:33

Euler's Method

Related Practice

Textbook Question

7–84. Evaluate the following integrals.

27. ∫ sin⁴(x/2) dx

Textbook Question

73. Two methods Evaluate ∫ dx/(x² - 1), for x > 1, in two ways: using partial fractions and a trigonometric substitution. Reconcile your two answers.

Textbook Question

54–57. {Use of Tech} Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with n = 4, 8, 16, and 32 subintervals when they are applied to the following integrals (with their exact values given).

59. ∫(from 0 to π) ln(5 + 3cosx) dx = π ln(9/2)

Textbook Question

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

53. ∫ (from 0 to 1) ln x dx

views

Textbook Question

Evaluate the following integrals.

∫ eˣ/(e²ˣ + 2eˣ + 17) dx

Textbook Question

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.

42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.