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

∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍

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Step 1: Recognize that the integral ∫ 2𝓍(𝓍² ― 1)⁹⁹ d𝓍 suggests a substitution method. Let u = 𝓍² ― 1, which simplifies the expression inside the parentheses.

Step 2: Compute the derivative of u with respect to 𝓍. Since u = 𝓍² ― 1, we find that du/d𝓍 = 2𝓍. Therefore, du = 2𝓍 d𝓍.

Step 3: Substitute u and du into the integral. Replace 𝓍² ― 1 with u and 2𝓍 d𝓍 with du. The integral becomes ∫ u⁹⁹ du.

Step 4: Apply the power rule for integration. Recall that ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C, where n ≠ -1. Using this rule, integrate u⁹⁹.

Step 5: Substitute back u = 𝓍² ― 1 into the result to express the solution in terms of 𝓍. Verify your work by differentiating the final expression 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 under curves and accumulation functions.

Change of Variables

The change of variables technique, also known as substitution, is a method 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 technique is particularly useful when dealing with complex expressions, allowing for easier evaluation of the integral.

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the antiderivative found corresponds to the original integrand. If the derivative of the integral result matches the integrand, it confirms that the integration was performed correctly, reinforcing the relationship between differentiation and integration.

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