Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 ∫ 𝓍eˣ² d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫𝓍eˣ² d𝓍 suggests a substitution method because the derivative of the inner function x² appears as a factor (𝓍). This is a common pattern for substitution.
2. Step 2: Let u = x². Then, compute the derivative of u with respect to x: du/dx = 2x, or equivalently, du = 2x dx.
3. Step 3: Rewrite the integral in terms of u. Substitute u = x² and du = 2x dx into the integral. This transforms the integral into (1/2)∫eᵘ du, where the factor of 1/2 comes from adjusting for the 2x in du.
4. Step 4: Evaluate the integral ∫eᵘ du. The antiderivative of eᵘ is simply eᵘ, so the integral becomes (1/2)eᵘ + C, where C is the constant of integration.
5. Step 5: Substitute back u = x² to express the result in terms of x. The final expression is (1/2)eˣ² + C. To verify, differentiate this result with respect to x and confirm that it matches the original integrand.

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, where the goal is to determine a function F(x) such that F'(x) equals the integrand.

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.

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 was performed accurately.