Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of sin(𝓍) multiplied by cos(𝓍). This suggests using a substitution method where u = sin(𝓍).
2. Step 2: Compute the derivative of u with respect to 𝓍: du/d𝓍 = cos(𝓍), which implies that du = cos(𝓍) d𝓍. Substitute u = sin(𝓍) and du = cos(𝓍) d𝓍 into the integral.
3. Step 3: Rewrite the integral in terms of u: ∫ (u⁵ + 3u³ - u) du. This simplifies the integral into a polynomial form.
4. Step 4: Apply the power rule for integration to each term of the polynomial: ∫ u⁵ du = (u⁶)/6, ∫ 3u³ du = (3u⁴)/4, and ∫ -u du = -(u²)/2.
5. Step 5: Combine the results to express the indefinite integral in terms of u, then substitute back u = sin(𝓍) to return to the original variable. Finally, check your work by differentiating the result to ensure it matches the original integrand.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives 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 essential for solving problems in calculus involving area under curves and accumulation of quantities.

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 functions or when the integrand can be expressed in terms of a simpler function.

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 original integrand is recovered, confirming that the integration was performed accurately. It serves as a crucial step in validating the solution and reinforcing the relationship between differentiation and integration.