Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ sec² (10𝓍 + 7) d𝓍

Verified step by step guidance

1. Step 1: Recognize the integral ∫ sec²(10𝓍 + 7) d𝓍 as a standard form that can be solved using a substitution method. The derivative of tan(u) is sec²(u), which suggests a substitution involving the argument of sec².
2. Step 2: Let u = 10𝓍 + 7. This substitution simplifies the argument of sec². Compute the derivative of u with respect to 𝓍: du/d𝓍 = 10, or equivalently, du = 10 d𝓍.
3. Step 3: Rewrite the integral in terms of u. Substitute u = 10𝓍 + 7 and du = 10 d𝓍 into the integral. The integral becomes ∫ sec²(u) * (1/10) du, where (1/10) is factored out.
4. Step 4: Evaluate the integral ∫ sec²(u) du using the standard result ∫ sec²(u) du = tan(u) + C, where C is the constant of integration.
5. Step 5: Substitute back u = 10𝓍 + 7 into the result to express the solution in terms of the original variable 𝓍. The final answer will be (1/10) * tan(10𝓍 + 7) + C.

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 essential for solving problems in calculus, particularly in finding areas under curves and solving differential equations.

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, 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 as inverse processes.