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

1:51 minutes

Problem 4.9.63

Textbook Question

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

∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1))dw

Verified step by step guidance

Step 1: Begin by analyzing the integrand ∫ ((e²ʷ - 5eʷ + 4)/(eʷ - 1)) dw. Notice that the numerator (e²ʷ - 5eʷ + 4) and denominator (eʷ - 1) suggest the possibility of polynomial division to simplify the fraction.

Step 2: Perform polynomial division. Divide the numerator (e²ʷ - 5eʷ + 4) by the denominator (eʷ - 1). This will yield a quotient and possibly a remainder. Write the integrand as the sum of the quotient and the remainder divided by the denominator.

Step 3: After simplifying, the integrand will be expressed as a sum of simpler terms. Break the integral into separate parts based on this decomposition. For example, ∫ (quotient) dw + ∫ (remainder/(eʷ - 1)) dw.

Step 4: Evaluate each term separately. For the quotient term, integrate directly. For the remainder term, consider substitution or other techniques if necessary. For example, if the remainder term involves (eʷ - 1), you might use substitution u = eʷ - 1.

Step 5: Combine the results of the individual integrals to write the final expression for the indefinite integral. Finally, check your work by differentiating the result to ensure it matches the original integrand.

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 often referred to as antiderivation, where one seeks a function whose derivative matches the given function.

Integration Techniques

To solve integrals, various techniques can be employed, such as substitution, integration by parts, or partial fraction decomposition. In this case, recognizing the structure of the integrand can help simplify the integral. For rational functions, breaking them down into simpler fractions can make integration more manageable.

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative. This process ensures that the derivative of the obtained function returns to the original integrand. This step is crucial for confirming the correctness of the integration process and solidifying understanding of the relationship between differentiation and integration.

Watch next

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

Start learning

Related Videos

Related Practice

Textbook Question