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

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1:59 minutes

Problem 8.1.1

Textbook Question

What change of variables would you use for the integral ∫(4 - 7x)^(-6) dx?

Verified step by step guidance

Step 1: Recognize that the integral involves a composite function (4 - 7x)^(-6). To simplify the integration, use a substitution method.

Step 2: Let u = 4 - 7x. This substitution simplifies the expression inside the integral. Compute the derivative of u with respect to x: du/dx = -7.

Step 3: Rearrange the derivative to express dx in terms of du: dx = du / (-7).

Step 4: Substitute u and dx into the integral. The integral becomes ∫u^(-6) * (du / -7).

Step 5: Factor out the constant -1/7 from the integral, leaving ∫u^(-6) du. This integral can now be solved using the power rule for integration.

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Key Concepts

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

Change of Variables

Change of variables, also known as substitution, is a technique used in integration to simplify the integral by transforming it into a more manageable form. This involves substituting a new variable for an expression in the integral, which can make the integration process easier. The goal is to find a relationship between the original variable and the new variable that simplifies the integrand.

Integration Techniques

Integration techniques are methods used to evaluate integrals that may not be solvable using basic antiderivatives. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for tackling complex integrals, as they provide strategies to break down the problem into simpler parts.

Power Rule for Integration

The power rule for integration states that the integral of x raised to the power n (where n ≠ -1) is (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule is fundamental in calculus and is often applied after performing a change of variables to simplify the integrand into a polynomial form, making it easier to integrate.