Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ (𝓍⁶ ― 3𝓍²)⁴ (𝓍⁵ ― 𝓍) d𝓍

Verified step by step guidance

1. Step 1: Identify a substitution that simplifies the integral. Notice that the term (𝓍⁶ ― 3𝓍²) appears raised to the fourth power, and its derivative is related to (𝓍⁵ ― 𝓍). Let u = 𝓍⁶ ― 3𝓍². Then, compute the derivative du/d𝓍 = 6𝓍⁵ ― 6𝓍.
2. Step 2: Rewrite the differential dx in terms of du. From du/d𝓍 = 6𝓍⁵ ― 6𝓍, we can solve for dx: dx = du / (6𝓍⁵ ― 6𝓍).
3. Step 3: Substitute u and dx into the integral. Replace (𝓍⁶ ― 3𝓍²) with u and (𝓍⁵ ― 𝓍) dx with du / (6𝓍⁵ ― 6𝓍). The integral becomes ∫ u⁴ du / (6𝓍⁵ ― 6𝓍).
4. Step 4: Simplify the integral. Notice that the term (𝓍⁵ ― 𝓍) cancels with the denominator (6𝓍⁵ ― 6𝓍) after substitution, leaving ∫ u⁴ du.
5. Step 5: Integrate u⁴ with respect to u. Use the power rule for integration: ∫ u⁴ du = (u⁵ / 5) + C, where C is the constant of integration. Finally, substitute back u = 𝓍⁶ ― 3𝓍² to express the result in terms of 𝓍.

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. 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.