Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ d𝓍 / (√1 ― 9𝓍²)

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Step 1: Recognize that the integral ∫ d𝓍 / (√1 ― 9𝓍²) resembles the standard form of an integral involving inverse trigonometric functions. Specifically, it matches the form ∫ dx / √(a² - x²), which corresponds to arcsin(x/a) + C.

Step 2: Identify the constants in the given integral. Here, a² = 1, so a = √1 = 1. Additionally, the term 9𝓍² can be rewritten as (3𝓍)², which suggests a substitution to simplify the integral.

Step 3: Perform a substitution to simplify the integral. Let u = 3𝓍, which implies that du = 3 d𝓍 or d𝓍 = du / 3. Substitute these into the integral to rewrite it in terms of u.

Step 4: After substitution, the integral becomes (1/3) ∫ du / √(1 - u²). This matches the standard form ∫ dx / √(a² - x²), where a = 1. The result of this integral is (1/3) arcsin(u/a) + C.

Step 5: Substitute back u = 3𝓍 into the result to express the solution in terms of the original variable 𝓍. The final answer is (1/3) arcsin(3𝓍) + C. Verify the solution by differentiating it to ensure it matches the original integrand.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, and it is fundamental in calculus for solving problems related to area, accumulation, and other applications.

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex expressions or when the integrand resembles a known derivative.

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be correct. This step is crucial in calculus as it ensures that the integration process was performed accurately.

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