Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              
                                                                                                                                                                                        
 ∫ (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

Verified step by step guidance

1. Step 1: Identify the substitution provided in the problem. Here, the substitution is u = 3𝓍² + 𝓍. Compute the derivative of u with respect to 𝓍: d𝓊/d𝓍 = 6𝓍 + 1.
2. Step 2: Rewrite the integral using the substitution. Replace √(3𝓍² + 𝓍) with √u and (6𝓍 + 1)d𝓍 with d𝓊, as d𝓊 = (6𝓍 + 1)d𝓍.
3. Step 3: The integral now becomes ∫ √u d𝓊. This is a simpler integral to evaluate.
4. Step 4: Use the power rule for integration to solve ∫ √u d𝓊. Recall that √u = u^(1/2), and the integral of u^(n) is (u^(n+1))/(n+1) + C, where C is the constant of integration.
5. Step 5: Substitute back u = 3𝓍² + 𝓍 into the result to express the solution in terms of 𝓍. Finally, check your answer by differentiating to ensure it matches the original integrand.

Key Concepts

Substitution in Integration

Substitution is a technique used in integration to simplify the integral by changing the variable of integration. By letting u be a function of x, we can express the integral in terms of u, making it easier to evaluate. The differential dx is also transformed according to the substitution, allowing us to rewrite the integral in a more manageable form.

Differentiation as a Check

Differentiation is the process of finding the derivative of a function, which can be used to verify the correctness of an integral. After evaluating an indefinite integral, differentiating the result should yield the original integrand. This serves as a crucial check to ensure that the integration process was performed correctly.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration (C) because the process of integration can yield multiple functions differing by a constant. Understanding the properties of indefinite integrals is essential for solving problems involving antiderivatives and applying techniques like substitution.