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

∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

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Step 1: Identify the substitution that simplifies the integral. Let u = √𝓍 + 1. Then, differentiate u with respect to 𝓍 to find du/d𝓍. Since u = √𝓍 + 1, we have du/d𝓍 = 1/(2√𝓍). Rearrange to express d𝓍 in terms of du: d𝓍 = 2√𝓍 du.

Step 2: Rewrite the integral in terms of u. Substitute √𝓍 + 1 with u and d𝓍 with 2√𝓍 du. The integral becomes ∫ [(u⁴) / (2√𝓍)] * (2√𝓍 du). Notice that the √𝓍 terms cancel out, simplifying the integral to ∫ u⁴ du.

Step 3: Integrate u⁴ with respect to u. Use the power rule for integration: ∫ u⁴ du = (u⁵)/5 + C, where C is the constant of integration.

Step 4: Substitute back the original variable 𝓍 into the solution. Recall that u = √𝓍 + 1, so replace u in the result with √𝓍 + 1. The integral becomes ((√𝓍 + 1)⁵)/5 + C.

Step 5: Verify the solution by differentiating the result. Differentiate ((√𝓍 + 1)⁵)/5 + C with respect to 𝓍 and confirm that it simplifies back to the original integrand [(√𝓍 + 1)⁴ / 2√𝓍]. This ensures the solution is correct.

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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

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, allowing for easier integration and ultimately leading to the correct antiderivative.

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 antiderivative matches the integrand, it confirms that the integration was performed correctly, reinforcing the relationship between differentiation and integration.

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