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

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Previous problemNext problem

2:13 minutes

Problem 4.9.61

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (eˣ⁺²) dx

Verified step by step guidance

Step 1: Recognize that the integral ∫(e^(x+2)) dx involves an exponential function. The general rule for integrating e^(u) is ∫e^(u) du = e^(u) + C, where u is a function of x.

Step 2: Identify the exponent in the given integral. Here, u = x + 2. To apply the rule, we need to check if the derivative of u with respect to x is present in the integral.

Step 3: Compute the derivative of u = x + 2 with respect to x. The derivative is du/dx = 1. Since the derivative is 1, we can directly integrate without additional adjustments.

Step 4: Apply the integration rule for e^(u). The integral becomes e^(x+2) + C, where C is the constant of integration.

Step 5: Verify your result by differentiating e^(x+2) + C. The derivative of e^(x+2) is e^(x+2), and the derivative of the constant C is 0. This confirms that the original integrand is recovered, ensuring the solution is correct.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

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 known as integration, which is the reverse operation of differentiation.

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. In the context of integration, the integral of e^(kx) is (1/k)e^(kx) + C, which is crucial for solving integrals involving exponential terms.

Differentiation Check

Checking work by differentiation involves taking the derivative of the obtained integral to verify that it matches the original integrand. This process ensures that the integration was performed correctly and helps identify any potential errors in the integration process.

Watch next

Master Introduction to Indefinite Integrals with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question