Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         
                                                                                                                                                                              
 ∫₀¹ 2e²ˣ d𝓍

Verified step by step guidance

1. Step 1: Recognize that the integral ∫₀¹ 2e²ˣ d𝓍 involves an exponential function. To simplify, use a substitution method. Let u = 2x, which implies du = 2 dx.
2. Step 2: Rewrite the integral in terms of u. Substitute dx = du/2 and adjust the limits of integration. When x = 0, u = 0; when x = 1, u = 2.
3. Step 3: The integral becomes ∫₀² eᵘ du after substitution. Notice that the constant factor 2 from the original integral is canceled by the 1/2 factor from dx = du/2.
4. Step 4: Use the formula for the integral of eᵘ, which is ∫ eᵘ du = eᵘ + C. Apply this formula to evaluate the integral ∫₀² eᵘ du.
5. Step 5: Evaluate the definite integral by substituting the limits of integration. Compute eᵘ at u = 2 and subtract eᵘ at u = 0 to find the result.

Key Concepts

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫_a^b f(x) dx, where f(x) is the function being integrated. The result of a definite integral is a number that quantifies the total accumulation of the function's values between the limits a and b.

Change of Variables

The change of variables technique, also known as substitution, is a method used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The Jacobian determinant is often used to adjust for the change in variable limits and the differential.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is the base of natural logarithms, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus, particularly in integration and differential equations. Understanding their properties is essential for evaluating integrals involving exponential terms.