Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ 𝓍³ (𝓍⁴ + 16)⁶ d𝓍

Verified step by step guidance

1. Step 1: Identify the substitution. Since the integral involves a composite function (𝓍⁴ + 16) raised to a power, let u = 𝓍⁴ + 16. This substitution simplifies the integral by reducing the complexity of the expression.
2. Step 2: Compute the derivative of u with respect to 𝓍. Differentiating u = 𝓍⁴ + 16 gives du/d𝓍 = 4𝓍³. Rearrange to express du in terms of d𝓍: du = 4𝓍³ d𝓍.
3. Step 3: Rewrite the integral in terms of u. Substitute u = 𝓍⁴ + 16 and du = 4𝓍³ d𝓍 into the original integral. The integral becomes (1/4) ∫ u⁶ du, where the factor of 1/4 accounts for the 4𝓍³ in du.
4. Step 4: Integrate with respect to u. Use the power rule for integration: ∫ u⁶ du = (u⁷)/7. Thus, the integral becomes (1/4) * (u⁷)/7.
5. Step 5: Substitute back u = 𝓍⁴ + 16 into the result. Replace u in the expression with the original variable to return to the original terms: (1/28) * (𝓍⁴ + 16)⁷ + C, where C is the constant of integration.

Key Concepts

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. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be accurate. This step is crucial in calculus as it ensures that the integration process was performed correctly and helps identify any potential errors in the calculations.