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

1:28 minutes

Problem 5.R.41

Textbook Question

Evaluating integrals Evaluate the following integrals.

∫ (9𝓍⁸―7𝓍⁶) d𝓍

Verified step by step guidance

Step 1: Recognize that the integral ∫ (9𝓍⁸ ― 7𝓍⁶) d𝓍 is a polynomial integral, which can be solved term by term using the power rule for integration.

Step 2: Apply the power rule for integration to the first term, 9𝓍⁸. The power rule states that ∫ 𝓍ⁿ d𝓍 = (𝓍ⁿ⁺¹)/(n+1) + C, where n is the exponent. For 9𝓍⁸, the integral becomes (9𝓍⁹)/9.

Step 3: Apply the power rule for integration to the second term, -7𝓍⁶. Using the same rule, the integral becomes (-7𝓍⁷)/7.

Step 4: Combine the results from Step 2 and Step 3 into a single expression. Remember to include the constant of integration, C, at the end.

Step 5: Simplify the coefficients in the combined expression to finalize the integral in its simplified form.

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.

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function on a given interval. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. The integral can be definite, with specific limits, or indefinite, representing a family of functions.

Power Rule for Integration

The Power Rule for Integration is a specific technique used to integrate polynomial functions. It states that the integral of x raised to the power n (where n is not equal to -1) is given by (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule simplifies the process of integrating polynomials by allowing for straightforward application to each term.

Constant Multiplication in Integration

When integrating a function that includes a constant multiplied by a variable term, the constant can be factored out of the integral. This means that if you have a function of the form k*f(x), where k is a constant, the integral can be expressed as k*∫f(x)dx. This property simplifies the integration process and allows for easier calculations.