Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ 𝓍 csc 𝓍² cot 𝓍² d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral involves a composite function and trigonometric identities. The term csc(𝓍²) cot(𝓍²) suggests a potential simplification using substitution.
2. Step 2: Let u = 𝓍². Then, differentiate u with respect to 𝓍 to find du: du = 2𝓍 d𝓍. This substitution will simplify the integral.
3. Step 3: Rewrite the integral in terms of u. Substitute 𝓍² with u and replace 𝓍 d𝓍 with (1/2) du. The integral becomes (1/2) ∫ csc(u) cot(u) du.
4. Step 4: Use the standard integral formula for ∫ csc(u) cot(u) du, which is -csc(u). This simplifies the integral to -(1/2) csc(u) + C, where C is the constant of integration.
5. Step 5: Substitute back u = 𝓍² to express the result in terms of the original variable. The final expression becomes -(1/2) csc(𝓍²) + C.

Key Concepts

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral involves determining the antiderivative of the function, which can often be done using various techniques such as substitution or integration by parts.

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 contains products or compositions of functions.

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 integration is confirmed to be accurate. This step is crucial in calculus as it ensures that the integration process has been performed correctly.