Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ sec 4w tan 4w dw

Verified step by step guidance

1. Step 1: Recognize the integral ∫ sec(4w) tan(4w) dw. This matches a standard form in calculus where the derivative of sec(x) is sec(x) tan(x).
2. Step 2: Use substitution to simplify the integral. Let u = 4w, which implies that du = 4 dw. Rewrite dw as dw = du/4.
3. Step 3: Substitute u into the integral. The integral becomes (1/4) ∫ sec(u) tan(u) du.
4. Step 4: Recall the standard result that the integral of sec(x) tan(x) is sec(x). Using this, the integral simplifies to (1/4) sec(u).
5. Step 5: Substitute back u = 4w to return to the original variable. The final expression is (1/4) sec(4w) + 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

The change of variables technique, also known as substitution, is a method 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 technique is particularly useful when dealing with composite functions or when the integrand contains products of functions that can be simplified.

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 correct. This step is crucial in calculus as it ensures that the integration process has been performed accurately and helps identify any potential errors in the calculation.