Use Table 5.6 to evaluate the following indefinite integrals.

(e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)

검증된 단계별 안내

Step 1: Start by factoring a 9 out of the denominator. Rewrite the integral as ∫ d𝓍 / (9(9 + 𝓍²)). This simplifies the denominator and prepares it for further manipulation.

Step 2: Factor out the constant 1/9 from the integral. This gives (1/9) ∫ d𝓍 / (9 + 𝓍²). Constants can be factored out of integrals to simplify calculations.

Step 3: Recognize the standard form of the integral. The denominator (9 + 𝓍²) matches the form a² + 𝓍², where a = 3. This suggests using the formula ∫ d𝓍 / (a² + 𝓍²) = (1/a) arctan(𝓍/a) + C.

Step 4: Apply the formula. Substitute a = 3 into the formula, resulting in (1/9) * (1/3) arctan(𝓍/3) + C. Combine the constants to simplify the expression further.

Step 5: Write the final simplified integral expression as (1/27) arctan(𝓍/3) + C. This is the indefinite integral of the given function.

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.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with the integral sign followed by the function and the differential, and they include a constant of integration, C. Understanding how to evaluate indefinite integrals is crucial for solving problems in calculus, as they provide the antiderivative of a function.

Factoring in Integration

Factoring is a technique used to simplify expressions, making them easier to integrate. In the context of the given problem, factoring a common term from the denominator can transform the integral into a more manageable form. This step is essential for applying integration techniques, such as substitution or recognizing standard integral forms.

Standard Integral Forms

Standard integral forms are well-known results that provide the antiderivatives of specific functions. These forms are often found in integral tables and can significantly expedite the integration process. Recognizing when an integral matches a standard form allows for quick evaluation, which is particularly useful in solving complex integrals efficiently.

Use Table 5.6 to evaluate the following indefinite integrals. (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)