Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ sin 𝓍 sec⁸ 𝓍 d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a combination of trigonometric functions, specifically sin(𝓍) and sec⁸(𝓍). To simplify, consider using a substitution method where u = sec(𝓍).
2. Step 2: Compute the derivative of u = sec(𝓍). Since the derivative of sec(𝓍) is sec(𝓍)tan(𝓍), we can express d𝓍 in terms of du: d𝓍 = du / (sec(𝓍)tan(𝓍)).
3. Step 3: Rewrite the integral ∫ sin(𝓍) sec⁸(𝓍) d𝓍 using the substitution u = sec(𝓍). Note that sin(𝓍) can be expressed as tan(𝓍)/sec(𝓍), which simplifies the integral further.
4. Step 4: Substitute all terms in the integral in terms of u. The integral becomes ∫ (tan(𝓍)/sec(𝓍)) sec⁸(𝓍) (du / (sec(𝓍)tan(𝓍))). Simplify the expression by canceling terms where possible.
5. Step 5: After simplification, the integral reduces to ∫ u⁷ du. Solve this integral using the power rule for integration: ∫ uⁿ du = (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Finally, substitute back u = sec(𝓍) 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 with the integral sign and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is also known as antiderivation, and it is essential for solving problems in calculus, particularly in finding areas under curves and solving differential equations.

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, making the integration process easier.

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 result matches the integrand, it confirms that the integration was performed correctly, providing a valuable verification step in solving calculus problems.