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 8.6.65

Chapter 8, Problem 8.6.65

Evaluate the following integrals.
65. ∫ from 0 to 1/6 1/√(1 - 9x²) dx

Verified step by step guidance

Step 1: Recognize the integral's structure. The integrand is of the form

1/√(1 - u²)

, which resembles the derivative of the arcsine function. Specifically, the derivative of

arcsin(u)

1/√(1 - u²)

Step 2: Identify the substitution needed to simplify the integral. Let

u = 3x

, which implies

du = 3 dx

. Rewrite the limits of integration accordingly: when

x = 0

u = 0

, and when

x = 1/6

u = 1/2

Step 3: Substitute into the integral. The integral becomes

∫ from 0 to 1/2 (1/3) * (1/√(1 - u²)) du

, where the factor

1/3

comes from adjusting for

du = 3 dx

Step 4: Apply the antiderivative of

1/√(1 - u²)

, which is

arcsin(u)

. The integral becomes

(1/3) * [arcsin(u)]

, evaluated from

u = 0

u = 1/2

Step 5: Substitute the limits of integration into the result. Compute

(1/3) * [arcsin(1/2) - arcsin(0)]

. Recall that

arcsin(1/2)

and

arcsin(0)

are standard values from trigonometry.

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.

Definite Integrals

A definite integral represents the signed area under a curve between two specified limits. It is denoted as ∫ from a to b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. The result of a definite integral is a numerical value that quantifies the accumulation of the function's values over the interval [a, b].

Substitution Method

The substitution method is a technique used to simplify the process of integration by changing the variable of integration. This involves substituting a new variable for a function of the original variable, which can make the integral easier to evaluate. It is particularly useful when dealing with integrals that involve composite functions or complicated expressions.

Trigonometric Substitution

Trigonometric substitution is a specific technique used in integration to simplify integrals involving square roots of quadratic expressions. By substituting a trigonometric function for a variable, such as x = (1/3)sin(θ) for √(1 - 9x²), the integral can often be transformed into a more manageable form. This method leverages the Pythagorean identities to facilitate the integration process.

Recommended video: