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

Verified step by step guidance

1. 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.
2. 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.
3. 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.
4. 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.
5. Step 5: Write the final simplified integral expression as (1/27) arctan(𝓍/3) + C. This is the indefinite integral of the given function.

Key Concepts

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.