Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

Ch. 8 - Integration Techniques

문제 8.6.65

8장, 문제 8.6.65

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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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