Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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2:33 minutes

Problem 5.5.46

Textbook Question

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

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.

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.

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.

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.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.